Rhonda numbers

From Rosetta Code
Task
Rhonda numbers
You are encouraged to solve this task according to the task description, using any language you may know.

A positive integer n is said to be a Rhonda number to base b if the product of the base b digits of n is equal to b times the sum of n's prime factors.


These numbers were named by Kevin Brown after an acquaintance of his whose residence number was 25662, a member of the base 10 numbers with this property.


25662 is a Rhonda number to base-10. The prime factorization is 2 × 3 × 7 × 13 × 47; the product of its base-10 digits is equal to the base times the sum of its prime factors:

2 × 5 × 6 × 6 × 2 = 720 = 10 × (2 + 3 + 7 + 13 + 47)

Rhonda numbers only exist in bases that are not a prime.

Rhonda numbers to base 10 always contain at least 1 digit 5 and always contain at least 1 even digit.


Task
  • For the non-prime bases b from 2 through 16 , find and display here, on this page, at least the first 10 Rhonda numbers to base b. Display the found numbers at least in base 10.


Stretch
  • Extend out to base 36.


See also



C++[edit]

#include <algorithm>
#include <cassert>
#include <iomanip>
#include <iostream>

int digit_product(int base, int n) {
    int product = 1;
    for (; n != 0; n /= base)
        product *= n % base;
    return product;
}

int prime_factor_sum(int n) {
    int sum = 0;
    for (; (n & 1) == 0; n >>= 1)
        sum += 2;
    for (int p = 3; p * p <= n; p += 2)
        for (; n % p == 0; n /= p)
            sum += p;
    if (n > 1)
        sum += n;
    return sum;
}

bool is_prime(int n) {
    if (n < 2)
        return false;
    if (n % 2 == 0)
        return n == 2;
    if (n % 3 == 0)
        return n == 3;
    for (int p = 5; p * p <= n; p += 4) {
        if (n % p == 0)
            return false;
        p += 2;
        if (n % p == 0)
            return false;
    }
    return true;
}

bool is_rhonda(int base, int n) {
    return digit_product(base, n) == base * prime_factor_sum(n);
}

std::string to_string(int base, int n) {
    assert(base <= 36);
    static constexpr char digits[] = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ";
    std::string str;
    for (; n != 0; n /= base)
        str += digits[n % base];
    std::reverse(str.begin(), str.end());
    return str;
}

int main() {
    const int limit = 15;
    for (int base = 2; base <= 36; ++base) {
        if (is_prime(base))
            continue;
        std::cout << "First " << limit << " Rhonda numbers to base " << base
                  << ":\n";
        int numbers[limit];
        for (int n = 1, count = 0; count < limit; ++n) {
            if (is_rhonda(base, n))
                numbers[count++] = n;
        }
        std::cout << "In base 10:";
        for (int i = 0; i < limit; ++i)
            std::cout << ' ' << numbers[i];
        std::cout << "\nIn base " << base << ':';
        for (int i = 0; i < limit; ++i)
            std::cout << ' ' << to_string(base, numbers[i]);
        std::cout << "\n\n";
    }
}
Output:
First 15 Rhonda numbers to base 4:
In base 10: 10206 11935 12150 16031 45030 94185 113022 114415 191149 244713 259753 374782 392121 503773 649902
In base 4: 2133132 2322133 2331312 3322133 22333212 112333221 123211332 123323233 232222231 323233221 333122221 1123133332 1133232321 1322333131 2132222232

First 15 Rhonda numbers to base 6:
In base 10: 855 1029 3813 5577 7040 7304 15104 19136 35350 36992 41031 42009 60368 65536 67821
In base 6: 3543 4433 25353 41453 52332 53452 153532 224332 431354 443132 513543 522253 1143252 1223224 1241553

First 15 Rhonda numbers to base 8:
In base 10: 1836 6318 6622 10530 14500 14739 17655 18550 25398 25956 30562 39215 39325 50875 51429
In base 8: 3454 14256 14736 24442 34244 34623 42367 44166 61466 62544 73542 114457 114635 143273 144345

First 15 Rhonda numbers to base 9:
In base 10: 15540 21054 25331 44360 44660 44733 47652 50560 54944 76857 77142 83334 83694 96448 97944
In base 9: 23276 31783 37665 66758 67232 67323 72326 76317 83328 126376 126733 136273 136723 156264 158316

First 15 Rhonda numbers to base 10:
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662

First 15 Rhonda numbers to base 12:
In base 10: 560 800 3993 4425 4602 4888 7315 8296 9315 11849 12028 13034 14828 15052 16264
In base 12: 3A8 568 2389 2689 27B6 29B4 4297 4974 5483 6A35 6B64 7662 86B8 8864 94B4

First 15 Rhonda numbers to base 14:
In base 10: 11475 18655 20565 29631 31725 45387 58404 58667 59950 63945 67525 68904 91245 99603 125543
In base 14: 4279 6B27 76CD AB27 B7C1 1277D 173DA 17547 17BC2 19437 1A873 1B17A 25377 28427 33A75

First 15 Rhonda numbers to base 15:
In base 10: 2392 2472 11468 15873 17424 18126 19152 20079 24388 30758 31150 33004 33550 37925 39483
In base 15: A97 AEC 35E8 4A83 5269 5586 5A1C 5E39 735D 91A8 936A 9BA4 9E1A B385 BA73

First 15 Rhonda numbers to base 16:
In base 10: 1000 1134 6776 15912 19624 20043 20355 23946 26296 29070 31906 32292 34236 34521 36465
In base 16: 3E8 46E 1A78 3E28 4CA8 4E4B 4F83 5D8A 66B8 718E 7CA2 7E24 85BC 86D9 8E71

First 15 Rhonda numbers to base 18:
In base 10: 1470 3000 8918 17025 19402 20650 21120 22156 26522 36549 38354 43281 46035 48768 54229
In base 18: 49C 94C 1998 2G9F 35FG 39D4 3B36 3E6G 49F8 64E9 6A6E 77A9 7G19 8696 956D

First 15 Rhonda numbers to base 20:
In base 10: 1815 11050 15295 21165 22165 30702 34510 34645 42292 44165 52059 53416 65945 78430 80712
In base 20: 4AF 17CA 1I4F 2CI5 2F85 3GF2 465A 46C5 55EC 5A85 6A2J 6DAG 84H5 9G1A A1FC

First 15 Rhonda numbers to base 21:
In base 10: 1632 5390 8512 12992 15678 25038 29412 34017 39552 48895 49147 61376 85078 89590 91798
In base 21: 3EF C4E J67 189E 1EBC 2EG6 33EC 3E2I 45E9 55I7 5697 6D3E 93J7 9E34 9J37

First 15 Rhonda numbers to base 22:
In base 10: 2695 4128 7865 28800 31710 37030 71875 74306 117760 117895 121626 126002 131427 175065 192753
In base 22: 5CB 8BE G5B 2FB2 2LB8 3AB4 6GB1 6LBC B16G B1CJ B96A BI78 C7BL G9FB I25B

First 15 Rhonda numbers to base 24:
In base 10: 2080 2709 3976 5628 5656 7144 8296 9030 10094 17612 20559 24616 26224 29106 31458
In base 24: 3EG 4GL 6LG 9IC 9JG C9G E9G FG6 HCE 16DK 1BGF 1IHG 1LCG 22CI 26EI

First 15 Rhonda numbers to base 25:
In base 10: 6764 9633 13260 22022 53382 57640 66015 69006 97014 140130 142880 144235 159724 162565 165504
In base 25: AKE FA8 L5A 1A5M 3AA7 3H5F 45FF 4AA6 655E 8O55 93F5 95JA A5DO AA2F AEK4

First 15 Rhonda numbers to base 26:
In base 10: 7788 9322 9374 11160 22165 27885 34905 44785 47385 49257 62517 72709 74217 108745 132302
In base 26: BDE DKE DME GD6 16KD 1F6D 1PGD 2E6D 2I2D 2KMD 3ECD 43ED 45KD 64MD 7DIE

First 15 Rhonda numbers to base 27:
In base 10: 4797 11844 12078 13200 14841 17750 24320 26883 27477 46455 52750 58581 61009 61446 61500
In base 27: 6FI G6I GF9 I2O K9I O9B 169K 19NI 1AII 29JF 2I9J 2Q9I 32IG 337L 339L

First 15 Rhonda numbers to base 28:
In base 10: 3094 5808 5832 7462 11160 13671 27270 28194 28638 39375 39550 49500 50862 52338 52938
In base 28: 3QE 7BC 7C8 9EE E6G HC7 16LQ 17QQ 18EM 1M67 1MCE 273O 28OE 2AL6 2BEI

First 15 Rhonda numbers to base 30:
In base 10: 3024 3168 5115 5346 5950 6762 7750 7956 8470 9476 9576 9849 10360 11495 13035
In base 30: 3AO 3FI 5KF 5S6 6IA 7FC 8IA 8P6 9CA AFQ AJ6 AS9 BFA CN5 EEF

First 15 Rhonda numbers to base 32:
In base 10: 1944 3600 13520 15876 16732 16849 25410 25752 28951 47472 49610 50968 61596 64904 74005
In base 32: 1SO 3GG D6G FG4 GAS GEH OQ2 P4O S8N 1EBG 1GEA 1HOO 1S4S 1VC8 288L

First 15 Rhonda numbers to base 33:
In base 10: 756 7040 7568 13826 24930 30613 59345 63555 64372 131427 227840 264044 313709 336385 344858
In base 33: MU 6FB 6VB CMW MTF S3M 1LGB 1PBU 1Q3M 3LML 6B78 7BFB 8O2B 9BTG 9JM8

First 15 Rhonda numbers to base 34:
In base 10: 5661 14161 15620 16473 22185 37145 125579 134692 135405 138472 140369 177086 250665 255552 295614
In base 34: 4UH C8H DHE E8H J6H W4H 36LH 3EHI 3F4H 3HQO 3JEH 4H6E 6CSH 6H28 7HOI

First 15 Rhonda numbers to base 35:
In base 10: 8232 9476 9633 18634 30954 41905 52215 52440 56889 61992 62146 66339 98260 102180 103305
In base 35: 6P7 7PQ 7U8 F7E P9E Y7A 17LU 17SA 1BFE 1FL7 1FPL 1J5E 2A7F 2DEF 2EBK

First 15 Rhonda numbers to base 36:
In base 10: 1000 4800 5670 8190 10998 12412 13300 15750 16821 23016 51612 52734 67744 70929 75030
In base 36: RS 3PC 4DI 6BI 8HI 9KS A9G C5I CZ9 HRC 13TO 14OU 1G9S 1IQ9 1LW6

Factor[edit]

Works with: Factor version 0.99 2022-04-03
USING: formatting grouping io kernel lists lists.lazy math
math.parser math.primes math.primes.factors prettyprint ranges
sequences sequences.extras ;

: rhonda? ( n base -- ? )
    [ [ >base 1 group ] keep '[ _ base> ] map-product ]
    [ swap factors sum * ] 2bi = ;

: rhonda ( base -- list ) 1 lfrom swap '[ _ rhonda? ] lfilter ;

: list. ( list base -- ) '[ _ >base write bl ] leach nl ;

:: rhonda. ( base -- )
    15 base rhonda ltake :> r
    base "First 15 Rhonda numbers to base %d:\n" printf
    "In base 10: " write r 10 list.
    base "In base %d: " printf r base list. ;

2 36 [a..b] [ prime? not ] filter [ rhonda. nl ] each
Output:
First 15 Rhonda numbers to base 4:
In base 10: 10206 11935 12150 16031 45030 94185 113022 114415 191149 244713 259753 374782 392121 503773 649902 
In base 4: 2133132 2322133 2331312 3322133 22333212 112333221 123211332 123323233 232222231 323233221 333122221 1123133332 1133232321 1322333131 2132222232 

First 15 Rhonda numbers to base 6:
In base 10: 855 1029 3813 5577 7040 7304 15104 19136 35350 36992 41031 42009 60368 65536 67821 
In base 6: 3543 4433 25353 41453 52332 53452 153532 224332 431354 443132 513543 522253 1143252 1223224 1241553 

First 15 Rhonda numbers to base 8:
In base 10: 1836 6318 6622 10530 14500 14739 17655 18550 25398 25956 30562 39215 39325 50875 51429 
In base 8: 3454 14256 14736 24442 34244 34623 42367 44166 61466 62544 73542 114457 114635 143273 144345 

First 15 Rhonda numbers to base 9:
In base 10: 15540 21054 25331 44360 44660 44733 47652 50560 54944 76857 77142 83334 83694 96448 97944 
In base 9: 23276 31783 37665 66758 67232 67323 72326 76317 83328 126376 126733 136273 136723 156264 158316 

First 15 Rhonda numbers to base 10:
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662 
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662 

First 15 Rhonda numbers to base 12:
In base 10: 560 800 3993 4425 4602 4888 7315 8296 9315 11849 12028 13034 14828 15052 16264 
In base 12: 3a8 568 2389 2689 27b6 29b4 4297 4974 5483 6a35 6b64 7662 86b8 8864 94b4 

First 15 Rhonda numbers to base 14:
In base 10: 11475 18655 20565 29631 31725 45387 58404 58667 59950 63945 67525 68904 91245 99603 125543 
In base 14: 4279 6b27 76cd ab27 b7c1 1277d 173da 17547 17bc2 19437 1a873 1b17a 25377 28427 33a75 

First 15 Rhonda numbers to base 15:
In base 10: 2392 2472 11468 15873 17424 18126 19152 20079 24388 30758 31150 33004 33550 37925 39483 
In base 15: a97 aec 35e8 4a83 5269 5586 5a1c 5e39 735d 91a8 936a 9ba4 9e1a b385 ba73 

First 15 Rhonda numbers to base 16:
In base 10: 1000 1134 6776 15912 19624 20043 20355 23946 26296 29070 31906 32292 34236 34521 36465 
In base 16: 3e8 46e 1a78 3e28 4ca8 4e4b 4f83 5d8a 66b8 718e 7ca2 7e24 85bc 86d9 8e71 

First 15 Rhonda numbers to base 18:
In base 10: 1470 3000 8918 17025 19402 20650 21120 22156 26522 36549 38354 43281 46035 48768 54229 
In base 18: 49c 94c 1998 2g9f 35fg 39d4 3b36 3e6g 49f8 64e9 6a6e 77a9 7g19 8696 956d 

First 15 Rhonda numbers to base 20:
In base 10: 1815 11050 15295 21165 22165 30702 34510 34645 42292 44165 52059 53416 65945 78430 80712 
In base 20: 4af 17ca 1i4f 2ci5 2f85 3gf2 465a 46c5 55ec 5a85 6a2j 6dag 84h5 9g1a a1fc 

First 15 Rhonda numbers to base 21:
In base 10: 1632 5390 8512 12992 15678 25038 29412 34017 39552 48895 49147 61376 85078 89590 91798 
In base 21: 3ef c4e j67 189e 1ebc 2eg6 33ec 3e2i 45e9 55i7 5697 6d3e 93j7 9e34 9j37 

First 15 Rhonda numbers to base 22:
In base 10: 2695 4128 7865 28800 31710 37030 71875 74306 117760 117895 121626 126002 131427 175065 192753 
In base 22: 5cb 8be g5b 2fb2 2lb8 3ab4 6gb1 6lbc b16g b1cj b96a bi78 c7bl g9fb i25b 

First 15 Rhonda numbers to base 24:
In base 10: 2080 2709 3976 5628 5656 7144 8296 9030 10094 17612 20559 24616 26224 29106 31458 
In base 24: 3eg 4gl 6lg 9ic 9jg c9g e9g fg6 hce 16dk 1bgf 1ihg 1lcg 22ci 26ei 

First 15 Rhonda numbers to base 25:
In base 10: 6764 9633 13260 22022 53382 57640 66015 69006 97014 140130 142880 144235 159724 162565 165504 
In base 25: ake fa8 l5a 1a5m 3aa7 3h5f 45ff 4aa6 655e 8o55 93f5 95ja a5do aa2f aek4 

First 15 Rhonda numbers to base 26:
In base 10: 7788 9322 9374 11160 22165 27885 34905 44785 47385 49257 62517 72709 74217 108745 132302 
In base 26: bde dke dme gd6 16kd 1f6d 1pgd 2e6d 2i2d 2kmd 3ecd 43ed 45kd 64md 7die 

First 15 Rhonda numbers to base 27:
In base 10: 4797 11844 12078 13200 14841 17750 24320 26883 27477 46455 52750 58581 61009 61446 61500 
In base 27: 6fi g6i gf9 i2o k9i o9b 169k 19ni 1aii 29jf 2i9j 2q9i 32ig 337l 339l 

First 15 Rhonda numbers to base 28:
In base 10: 3094 5808 5832 7462 11160 13671 27270 28194 28638 39375 39550 49500 50862 52338 52938 
In base 28: 3qe 7bc 7c8 9ee e6g hc7 16lq 17qq 18em 1m67 1mce 273o 28oe 2al6 2bei 

First 15 Rhonda numbers to base 30:
In base 10: 3024 3168 5115 5346 5950 6762 7750 7956 8470 9476 9576 9849 10360 11495 13035 
In base 30: 3ao 3fi 5kf 5s6 6ia 7fc 8ia 8p6 9ca afq aj6 as9 bfa cn5 eef 

First 15 Rhonda numbers to base 32:
In base 10: 1944 3600 13520 15876 16732 16849 25410 25752 28951 47472 49610 50968 61596 64904 74005 
In base 32: 1so 3gg d6g fg4 gas geh oq2 p4o s8n 1ebg 1gea 1hoo 1s4s 1vc8 288l 

First 15 Rhonda numbers to base 33:
In base 10: 756 7040 7568 13826 24930 30613 59345 63555 64372 131427 227840 264044 313709 336385 344858 
In base 33: mu 6fb 6vb cmw mtf s3m 1lgb 1pbu 1q3m 3lml 6b78 7bfb 8o2b 9btg 9jm8 

First 15 Rhonda numbers to base 34:
In base 10: 5661 14161 15620 16473 22185 37145 125579 134692 135405 138472 140369 177086 250665 255552 295614 
In base 34: 4uh c8h dhe e8h j6h w4h 36lh 3ehi 3f4h 3hqo 3jeh 4h6e 6csh 6h28 7hoi 

First 15 Rhonda numbers to base 35:
In base 10: 8232 9476 9633 18634 30954 41905 52215 52440 56889 61992 62146 66339 98260 102180 103305 
In base 35: 6p7 7pq 7u8 f7e p9e y7a 17lu 17sa 1bfe 1fl7 1fpl 1j5e 2a7f 2def 2ebk 

First 15 Rhonda numbers to base 36:
In base 10: 1000 4800 5670 8190 10998 12412 13300 15750 16821 23016 51612 52734 67744 70929 75030 
In base 36: rs 3pc 4di 6bi 8hi 9ks a9g c5i cz9 hrc 13to 14ou 1g9s 1iq9 1lw6 

Go[edit]

Translation of: Wren
Library: Go-rcu
package main

import (
    "fmt"
    "rcu"
    "strconv"
)

func contains(a []int, n int) bool {
    for _, e := range a {
        if e == n {
            return true
        }
    }
    return false
}

func main() {
    for b := 2; b <= 36; b++ {
        if rcu.IsPrime(b) {
            continue
        }
        count := 0
        var rhonda []int
        for n := 1; count < 15; n++ {
            digits := rcu.Digits(n, b)
            if !contains(digits, 0) {
                var anyEven = false
                for _, d := range digits {
                    if d%2 == 0 {
                        anyEven = true
                        break
                    }
                }
                if b != 10 || (contains(digits, 5) && anyEven) {
                    calc1 := 1
                    for _, d := range digits {
                        calc1 *= d
                    }
                    calc2 := b * rcu.SumInts(rcu.PrimeFactors(n))
                    if calc1 == calc2 {
                        rhonda = append(rhonda, n)
                        count++
                    }
                }
            }
        }
        if len(rhonda) > 0 {
            fmt.Printf("\nFirst 15 Rhonda numbers in base %d:\n", b)
            rhonda2 := make([]string, len(rhonda))
            counts2 := make([]int, len(rhonda))
            for i, r := range rhonda {
                rhonda2[i] = fmt.Sprintf("%d", r)
                counts2[i] = len(rhonda2[i])
            }
            rhonda3 := make([]string, len(rhonda))
            counts3 := make([]int, len(rhonda))
            for i, r := range rhonda {
                rhonda3[i] = strconv.FormatInt(int64(r), b)
                counts3[i] = len(rhonda3[i])
            }
            maxLen2 := rcu.MaxInts(counts2)
            maxLen3 := rcu.MaxInts(counts3)
            maxLen := maxLen2
            if maxLen3 > maxLen {
                maxLen = maxLen3
            }
            maxLen++
            fmt.Printf("In base 10: %*s\n", maxLen, rhonda2)
            fmt.Printf("In base %-2d: %*s\n", b, maxLen, rhonda3)
        }
    }
}
Output:
First 15 Rhonda numbers in base 4:
In base 10: [      10206       11935       12150       16031       45030       94185      113022      114415      191149      244713      259753      374782      392121      503773      649902]
In base 4 : [    2133132     2322133     2331312     3322133    22333212   112333221   123211332   123323233   232222231   323233221   333122221  1123133332  1133232321  1322333131  2132222232]

First 15 Rhonda numbers in base 6:
In base 10: [     855     1029     3813     5577     7040     7304    15104    19136    35350    36992    41031    42009    60368    65536    67821]
In base 6 : [    3543     4433    25353    41453    52332    53452   153532   224332   431354   443132   513543   522253  1143252  1223224  1241553]

First 15 Rhonda numbers in base 8:
In base 10: [   1836    6318    6622   10530   14500   14739   17655   18550   25398   25956   30562   39215   39325   50875   51429]
In base 8 : [   3454   14256   14736   24442   34244   34623   42367   44166   61466   62544   73542  114457  114635  143273  144345]

First 15 Rhonda numbers in base 9:
In base 10: [  15540   21054   25331   44360   44660   44733   47652   50560   54944   76857   77142   83334   83694   96448   97944]
In base 9 : [  23276   31783   37665   66758   67232   67323   72326   76317   83328  126376  126733  136273  136723  156264  158316]

First 15 Rhonda numbers in base 10:
In base 10: [  1568   2835   4752   5265   5439   5664   5824   5832   8526  12985  15625  15698  19435  25284  25662]
In base 10: [  1568   2835   4752   5265   5439   5664   5824   5832   8526  12985  15625  15698  19435  25284  25662]

First 15 Rhonda numbers in base 12:
In base 10: [   560    800   3993   4425   4602   4888   7315   8296   9315  11849  12028  13034  14828  15052  16264]
In base 12: [   3a8    568   2389   2689   27b6   29b4   4297   4974   5483   6a35   6b64   7662   86b8   8864   94b4]

First 15 Rhonda numbers in base 14:
In base 10: [  11475   18655   20565   29631   31725   45387   58404   58667   59950   63945   67525   68904   91245   99603  125543]
In base 14: [   4279    6b27    76cd    ab27    b7c1   1277d   173da   17547   17bc2   19437   1a873   1b17a   25377   28427   33a75]

First 15 Rhonda numbers in base 15:
In base 10: [  2392   2472  11468  15873  17424  18126  19152  20079  24388  30758  31150  33004  33550  37925  39483]
In base 15: [   a97    aec   35e8   4a83   5269   5586   5a1c   5e39   735d   91a8   936a   9ba4   9e1a   b385   ba73]

First 15 Rhonda numbers in base 16:
In base 10: [  1000   1134   6776  15912  19624  20043  20355  23946  26296  29070  31906  32292  34236  34521  36465]
In base 16: [   3e8    46e   1a78   3e28   4ca8   4e4b   4f83   5d8a   66b8   718e   7ca2   7e24   85bc   86d9   8e71]

First 15 Rhonda numbers in base 18:
In base 10: [  1470   3000   8918  17025  19402  20650  21120  22156  26522  36549  38354  43281  46035  48768  54229]
In base 18: [   49c    94c   1998   2g9f   35fg   39d4   3b36   3e6g   49f8   64e9   6a6e   77a9   7g19   8696   956d]

First 15 Rhonda numbers in base 20:
In base 10: [  1815  11050  15295  21165  22165  30702  34510  34645  42292  44165  52059  53416  65945  78430  80712]
In base 20: [   4af   17ca   1i4f   2ci5   2f85   3gf2   465a   46c5   55ec   5a85   6a2j   6dag   84h5   9g1a   a1fc]

First 15 Rhonda numbers in base 21:
In base 10: [  1632   5390   8512  12992  15678  25038  29412  34017  39552  48895  49147  61376  85078  89590  91798]
In base 21: [   3ef    c4e    j67   189e   1ebc   2eg6   33ec   3e2i   45e9   55i7   5697   6d3e   93j7   9e34   9j37]

First 15 Rhonda numbers in base 22:
In base 10: [   2695    4128    7865   28800   31710   37030   71875   74306  117760  117895  121626  126002  131427  175065  192753]
In base 22: [    5cb     8be     g5b    2fb2    2lb8    3ab4    6gb1    6lbc    b16g    b1cj    b96a    bi78    c7bl    g9fb    i25b]

First 15 Rhonda numbers in base 24:
In base 10: [  2080   2709   3976   5628   5656   7144   8296   9030  10094  17612  20559  24616  26224  29106  31458]
In base 24: [   3eg    4gl    6lg    9ic    9jg    c9g    e9g    fg6    hce   16dk   1bgf   1ihg   1lcg   22ci   26ei]

First 15 Rhonda numbers in base 25:
In base 10: [   6764    9633   13260   22022   53382   57640   66015   69006   97014  140130  142880  144235  159724  162565  165504]
In base 25: [    ake     fa8     l5a    1a5m    3aa7    3h5f    45ff    4aa6    655e    8o55    93f5    95ja    a5do    aa2f    aek4]

First 15 Rhonda numbers in base 26:
In base 10: [   7788    9322    9374   11160   22165   27885   34905   44785   47385   49257   62517   72709   74217  108745  132302]
In base 26: [    bde     dke     dme     gd6    16kd    1f6d    1pgd    2e6d    2i2d    2kmd    3ecd    43ed    45kd    64md    7die]

First 15 Rhonda numbers in base 27:
In base 10: [  4797  11844  12078  13200  14841  17750  24320  26883  27477  46455  52750  58581  61009  61446  61500]
In base 27: [   6fi    g6i    gf9    i2o    k9i    o9b   169k   19ni   1aii   29jf   2i9j   2q9i   32ig   337l   339l]

First 15 Rhonda numbers in base 28:
In base 10: [  3094   5808   5832   7462  11160  13671  27270  28194  28638  39375  39550  49500  50862  52338  52938]
In base 28: [   3qe    7bc    7c8    9ee    e6g    hc7   16lq   17qq   18em   1m67   1mce   273o   28oe   2al6   2bei]

First 15 Rhonda numbers in base 30:
In base 10: [  3024   3168   5115   5346   5950   6762   7750   7956   8470   9476   9576   9849  10360  11495  13035]
In base 30: [   3ao    3fi    5kf    5s6    6ia    7fc    8ia    8p6    9ca    afq    aj6    as9    bfa    cn5    eef]

First 15 Rhonda numbers in base 32:
In base 10: [  1944   3600  13520  15876  16732  16849  25410  25752  28951  47472  49610  50968  61596  64904  74005]
In base 32: [   1so    3gg    d6g    fg4    gas    geh    oq2    p4o    s8n   1ebg   1gea   1hoo   1s4s   1vc8   288l]

First 15 Rhonda numbers in base 33:
In base 10: [    756    7040    7568   13826   24930   30613   59345   63555   64372  131427  227840  264044  313709  336385  344858]
In base 33: [     mu     6fb     6vb     cmw     mtf     s3m    1lgb    1pbu    1q3m    3lml    6b78    7bfb    8o2b    9btg    9jm8]

First 15 Rhonda numbers in base 34:
In base 10: [   5661   14161   15620   16473   22185   37145  125579  134692  135405  138472  140369  177086  250665  255552  295614]
In base 34: [    4uh     c8h     dhe     e8h     j6h     w4h    36lh    3ehi    3f4h    3hqo    3jeh    4h6e    6csh    6h28    7hoi]

First 15 Rhonda numbers in base 35:
In base 10: [   8232    9476    9633   18634   30954   41905   52215   52440   56889   61992   62146   66339   98260  102180  103305]
In base 35: [    6p7     7pq     7u8     f7e     p9e     y7a    17lu    17sa    1bfe    1fl7    1fpl    1j5e    2a7f    2def    2ebk]

First 15 Rhonda numbers in base 36:
In base 10: [  1000   4800   5670   8190  10998  12412  13300  15750  16821  23016  51612  52734  67744  70929  75030]
In base 36: [    rs    3pc    4di    6bi    8hi    9ks    a9g    c5i    cz9    hrc   13to   14ou   1g9s   1iq9   1lw6]

J[edit]

tobase=: (a.{~;48 97(+ i.)each 10 26) {~ #.inv
isrhonda=: (*/@:(#.inv) = (* +/@q:))"0

task=: {{
  for_base.(#~ 0=1&p:) }.1+i.36 do.
    k=.i.0
    block=. 1+i.1e4
    while. 15>#k do.
      k=. k, block#~ base isrhonda block
      block=. block+1e4
    end.
    echo ''
    echo 'First 15 Rhondas in',b=.' base ',':',~":base
    echo 'In base 10: ',":15{.k
    echo 'In',;:inv b;base tobase each 15{.k
  end.
}}

   task''
Output:
First 15 Rhondas in base 4:
In base 10: 10206 11935 12150 16031 45030 94185 113022 114415 191149 244713 259753 374782 392121 503773 649902
In base 4: 2133132 2322133 2331312 3322133 22333212 112333221 123211332 123323233 232222231 323233221 333122221 1123133332 1133232321 1322333131 2132222232

First 15 Rhondas in base 6:
In base 10: 855 1029 3813 5577 7040 7304 15104 19136 35350 36992 41031 42009 60368 65536 67821
In base 6: 3543 4433 25353 41453 52332 53452 153532 224332 431354 443132 513543 522253 1143252 1223224 1241553

First 15 Rhondas in base 8:
In base 10: 1836 6318 6622 10530 14500 14739 17655 18550 25398 25956 30562 39215 39325 50875 51429
In base 8: 3454 14256 14736 24442 34244 34623 42367 44166 61466 62544 73542 114457 114635 143273 144345

First 15 Rhondas in base 9:
In base 10: 15540 21054 25331 44360 44660 44733 47652 50560 54944 76857 77142 83334 83694 96448 97944
In base 9: 23276 31783 37665 66758 67232 67323 72326 76317 83328 126376 126733 136273 136723 156264 158316

First 15 Rhondas in base 10:
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662

First 15 Rhondas in base 12:
In base 10: 560 800 3993 4425 4602 4888 7315 8296 9315 11849 12028 13034 14828 15052 16264
In base 12: 3a8 568 2389 2689 27b6 29b4 4297 4974 5483 6a35 6b64 7662 86b8 8864 94b4

First 15 Rhondas in base 14:
In base 10: 11475 18655 20565 29631 31725 45387 58404 58667 59950 63945 67525 68904 91245 99603 125543
In base 14: 4279 6b27 76cd ab27 b7c1 1277d 173da 17547 17bc2 19437 1a873 1b17a 25377 28427 33a75

First 15 Rhondas in base 15:
In base 10: 2392 2472 11468 15873 17424 18126 19152 20079 24388 30758 31150 33004 33550 37925 39483
In base 15: a97 aec 35e8 4a83 5269 5586 5a1c 5e39 735d 91a8 936a 9ba4 9e1a b385 ba73

First 15 Rhondas in base 16:
In base 10: 1000 1134 6776 15912 19624 20043 20355 23946 26296 29070 31906 32292 34236 34521 36465
In base 16: 3e8 46e 1a78 3e28 4ca8 4e4b 4f83 5d8a 66b8 718e 7ca2 7e24 85bc 86d9 8e71

First 15 Rhondas in base 18:
In base 10: 1470 3000 8918 17025 19402 20650 21120 22156 26522 36549 38354 43281 46035 48768 54229
In base 18: 49c 94c 1998 2g9f 35fg 39d4 3b36 3e6g 49f8 64e9 6a6e 77a9 7g19 8696 956d

First 15 Rhondas in base 20:
In base 10: 1815 11050 15295 21165 22165 30702 34510 34645 42292 44165 52059 53416 65945 78430 80712
In base 20: 4af 17ca 1i4f 2ci5 2f85 3gf2 465a 46c5 55ec 5a85 6a2j 6dag 84h5 9g1a a1fc

First 15 Rhondas in base 21:
In base 10: 1632 5390 8512 12992 15678 25038 29412 34017 39552 48895 49147 61376 85078 89590 91798
In base 21: 3ef c4e j67 189e 1ebc 2eg6 33ec 3e2i 45e9 55i7 5697 6d3e 93j7 9e34 9j37

First 15 Rhondas in base 22:
In base 10: 2695 4128 7865 28800 31710 37030 71875 74306 117760 117895 121626 126002 131427 175065 192753
In base 22: 5cb 8be g5b 2fb2 2lb8 3ab4 6gb1 6lbc b16g b1cj b96a bi78 c7bl g9fb i25b

First 15 Rhondas in base 24:
In base 10: 2080 2709 3976 5628 5656 7144 8296 9030 10094 17612 20559 24616 26224 29106 31458
In base 24: 3eg 4gl 6lg 9ic 9jg c9g e9g fg6 hce 16dk 1bgf 1ihg 1lcg 22ci 26ei

First 15 Rhondas in base 25:
In base 10: 6764 9633 13260 22022 53382 57640 66015 69006 97014 140130 142880 144235 159724 162565 165504
In base 25: ake fa8 l5a 1a5m 3aa7 3h5f 45ff 4aa6 655e 8o55 93f5 95ja a5do aa2f aek4

First 15 Rhondas in base 26:
In base 10: 7788 9322 9374 11160 22165 27885 34905 44785 47385 49257 62517 72709 74217 108745 132302
In base 26: bde dke dme gd6 16kd 1f6d 1pgd 2e6d 2i2d 2kmd 3ecd 43ed 45kd 64md 7die

First 15 Rhondas in base 27:
In base 10: 4797 11844 12078 13200 14841 17750 24320 26883 27477 46455 52750 58581 61009 61446 61500
In base 27: 6fi g6i gf9 i2o k9i o9b 169k 19ni 1aii 29jf 2i9j 2q9i 32ig 337l 339l

First 15 Rhondas in base 28:
In base 10: 3094 5808 5832 7462 11160 13671 27270 28194 28638 39375 39550 49500 50862 52338 52938
In base 28: 3qe 7bc 7c8 9ee e6g hc7 16lq 17qq 18em 1m67 1mce 273o 28oe 2al6 2bei

First 15 Rhondas in base 30:
In base 10: 3024 3168 5115 5346 5950 6762 7750 7956 8470 9476 9576 9849 10360 11495 13035
In base 30: 3ao 3fi 5kf 5s6 6ia 7fc 8ia 8p6 9ca afq aj6 as9 bfa cn5 eef

First 15 Rhondas in base 32:
In base 10: 1944 3600 13520 15876 16732 16849 25410 25752 28951 47472 49610 50968 61596 64904 74005
In base 32: 1so 3gg d6g fg4 gas geh oq2 p4o s8n 1ebg 1gea 1hoo 1s4s 1vc8 288l

First 15 Rhondas in base 33:
In base 10: 756 7040 7568 13826 24930 30613 59345 63555 64372 131427 227840 264044 313709 336385 344858
In base 33: mu 6fb 6vb cmw mtf s3m 1lgb 1pbu 1q3m 3lml 6b78 7bfb 8o2b 9btg 9jm8

First 15 Rhondas in base 34:
In base 10: 5661 14161 15620 16473 22185 37145 125579 134692 135405 138472 140369 177086 250665 255552 295614
In base 34: 4uh c8h dhe e8h j6h w4h 36lh 3ehi 3f4h 3hqo 3jeh 4h6e 6csh 6h28 7hoi

First 15 Rhondas in base 35:
In base 10: 8232 9476 9633 18634 30954 41905 52215 52440 56889 61992 62146 66339 98260 102180 103305
In base 35: 6p7 7pq 7u8 f7e p9e y7a 17lu 17sa 1bfe 1fl7 1fpl 1j5e 2a7f 2def 2ebk

First 15 Rhondas in base 36:
In base 10: 1000 4800 5670 8190 10998 12412 13300 15750 16821 23016 51612 52734 67744 70929 75030
In base 36: rs 3pc 4di 6bi 8hi 9ks a9g c5i cz9 hrc 13to 14ou 1g9s 1iq9 1lw6

Hoon[edit]

Library file (e.g. /lib/rhonda.hoon):

::
::  A library for producing Rhonda numbers and testing if numbers are Rhonda.
::
::    A number is Rhonda if the product of its digits of in base b equals 
::    the product of the base b and the sum of its prime factors.
::    see also: https://mathworld.wolfram.com/RhondaNumber.html
::
=<
::
|%
::  +check: test whether the number n is Rhonda to base b
::
++  check
  |=  [b=@ud n=@ud]
  ^-  ?
  ~_  leaf+"base b must be >= 2"
  ?>  (gte b 2)
  ~_  leaf+"candidate number n must be >= 2"
  ?>  (gte n 2)
  ::
  .=  (roll (base-digits b n) mul)
  %+  mul
    b
  (roll (prime-factors n) add)
::  +series: produce the first n numbers which are Rhonda in base b
::
::    produce ~ if base b has no Rhonda numbers
::
++  series
  |=  [b=@ud n=@ud]
  ^-  (list @ud)
  ~_  leaf+"base b must be >= 2"
  ?>  (gte b 2)
  ::
  ?:  =((prime-factors b) ~[b])
    ~
  =/  candidate=@ud  2
  =+  rhondas=*(list @ud)
  |-
  ?:  =(n 0)
    (flop rhondas)
  =/  is-rhonda=?  (check b candidate)
  %=  $
    rhondas    ?:(is-rhonda [candidate rhondas] rhondas)
    n          ?:(is-rhonda (dec n) n)
    candidate  +(candidate)
  ==
--
::
|%
::  +base-digits: produce a list of the digits of n represented in base b
::
::    This arm has two behaviors which may be at first surprising, but do not
::    matter for the purposes of the ++check and ++series arms, and allow for
::    some simplifications to its implementation.
::    - crashes on n=0
::    - orders the list of digits with least significant digits first
::
::    ex: (base-digits 4 10.206) produces ~[2 3 1 3 3 1 2]
::
++  base-digits
  |=  [b=@ud n=@ud]
  ^-  (list @ud)
  ?>  (gte b 2)
  ?<  =(n 0)
  ::
  |-
  ?:  =(n 0)
    ~
  :-  (mod n b)
  $(n (div n b))
::  +prime-factors: produce a list of the prime factors of n
::    
::    by trial division
::    n must be >= 2
::    if n is prime, produce ~[n]
::    ex: (prime-factors 10.206) produces ~[7 3 3 3 3 3 3 2]
::
++  prime-factors
  |=  [n=@ud]
  ^-  (list @ud)
  ?>  (gte n 2)
  ::
  =+  factors=*(list @ud)
  =/  wheel  new-wheel
  ::  test candidates as produced by the wheel, not exceeding sqrt(n) 
  ::
  |-
  =^  candidate  wheel  (next:wheel)
  ?.  (lte (mul candidate candidate) n)
    ?:((gth n 1) [n factors] factors)
  |-
  ?:  =((mod n candidate) 0)
    ::  repeat the prime factor as many times as possible
    ::
    $(factors [candidate factors], n (div n candidate))
  ^$
::  +new-wheel: a door for generating numbers that may be prime
::
::    This uses wheel factorization with a basis of {2, 3, 5} to limit the
::    number of composites produced. It produces numbers in increasing order
::    starting from 2.
::
++  new-wheel
  =/  fixed=(list @ud)  ~[2 3 5 7]
  =/  skips=(list @ud)  ~[4 2 4 2 4 6 2 6]
  =/  lent-fixed=@ud  (lent fixed)
  =/  lent-skips=@ud  (lent skips)
  ::
  |_  [current=@ud fixed-i=@ud skips-i=@ud]
  ::  +next: produce the next number and the new wheel state
  ::
  ++  next
    |.
    ::  Exhaust the numbers in fixed. Then calculate successive values by
    ::  cycling through skips and increasing from the previous number by
    ::  the current skip-value.
    ::
    =/  fixed-done=?  =(fixed-i lent-fixed)
    =/  next-fixed-i  ?:(fixed-done fixed-i +(fixed-i))
    =/  next-skips-i  ?:(fixed-done (mod +(skips-i) lent-skips) skips-i)
    =/  next
    ?.  fixed-done
      (snag fixed-i fixed)
    (add current (snag skips-i skips))
    :-  next
    +.$(current next, fixed-i next-fixed-i, skips-i next-skips-i)
  --
--

Script file ("generator") (e.g. /gen/rhonda.hoon):

/+  *rhonda
:-  %say
|=  [* [base=@ud many=@ud ~] ~]
:-  %noun
(series base many)

Alternative library file using map (associative array):

|%
++  check
  |=  [n=@ud base=@ud]
  ::  if base is prime, automatic no
  ::
  ?:  =((~(gut by (prime-map +(base))) base 0) 0)
    %.n
  ::  if not multiply the digits and compare to base x sum of factors
  ::
  ?:  =((roll (digits [base n]) mul) (mul base (roll (factor n) add)))
    %.y
  %.n
++  series
  |=  [base=@ud many=@ud]
  =/  rhondas  *(list @ud)
  ?:  =((~(gut by (prime-map +(base))) base 0) 0)
    rhondas
  =/  itr  1
  |-
  ?:  =((lent rhondas) many)
    (flop rhondas)
  ?:  =((check itr base) %.n)
    $(itr +(itr))
  $(rhondas [itr rhondas], itr +(itr))
::  digits: gives the list of digits of a number in a base
::
::    We strip digits least to most significant.
::    The least significant digit (lsd) of n in base b is just n mod b.
::    Subtract the lsd, divide by b, and repeat.
::    To know when to stop, we need to know how many digits there are.
++  digits
  |=  [base=@ud num=@ud]
  ^-  (list @ud)
  |-
  =/  modulus=@ud  (mod num base)
  ?:  =((num-digits base num) 1)
    ~[modulus]
  [modulus $(num (div (sub num modulus) base))]
::  num-digits: gives the number of digits of a number in a base
::
::    Simple idea: k is the number of digits of n in base b if and
::    only if k is the smallest number such that b^k > n.
++  num-digits
  |=  [base=@ud num=@ud]
  ^-  @ud
  =/  digits=@ud  1
  |-
  ?:  (gth (pow base digits) num)
    digits
  $(digits +(digits))
::  factor: produce a list of prime factors
::
::    The idea is to identify "small factors" of n, i.e. prime factors less than
::    the square root. We then divide n by these factors to reduce the
::    magnitude of n. It's easy to argue that after this is done, we obtain 1
::    or the largest prime factor.
::
++  factor
  |=  n=@ud
  ^-  (list @ud)
  ?:  ?|(=(n 0) =(n 1))
    ~[n]
  =/  factorization  *(list @ud)
  ::  produce primes less than or equal to root n
  ::
  =/  root  (sqrt n)
  =/  primes  (prime-map +(root))
  ::  itr = iterate; we want to iterate through the primes less than root n
  ::
  =/  itr  2
  |-
  ?:  =(itr +(root))
  ::  if n is now 1 we're done
  ::
    ?:  =(n 1)
      factorization
    ::  otherwise it's now the original n's largest primes factor
    ::
    [n factorization]
  ::  if itr not prime move on
  ::
  ?:  =((~(gut by primes) itr 0) 1)
    $(itr +(itr))
  ::  if it is prime, divide out by the highest power that divides num
  ::
  ?:  =((mod n itr) 0)
    $(n (div n itr), factorization [itr factorization])
  ::  once done, move to next prime
  ::
  $(itr +(itr))
::  sqrt: gives the integer square root of a number
::
::    It's based on an algorithm that predates the Greeks:
::    To find the square root of A, think of A as an area.
::    Guess the side of the square x. Compute the other side y = A/x.
::    If x is an over/underestimate then y is an under/overestimate.
::    So (x+y)/2 is the average of an over and underestimate, thus better than x.
::    Repeatedly doing x --> (x + A/x)/2 converges to sqrt(A).
::
::    This algorithm is the same but with integer valued operations.
::    The algorithm either converges to the integer square root and repeats,
::    or gets trapped in a two-cycle of adjacent integers.
::    In the latter case, the smaller number is the answer.
::
++  sqrt
  |=  n=@ud
  =/  guess=@ud  1
  |-
  =/  new-guess  (div (add guess (div n guess)) 2)
  ::  sequence stabilizes
  ::
  ?:  =(guess new-guess)
    guess
  ::  sequence is trapped in 2-cycle
  ::
  ?:  =(guess +(new-guess))
    new-guess
  ?:  =(new-guess +(guess))
    guess
  $(guess new-guess)
::  prime-map: (effectively) produces primes less than a given input
::
::    This is the sieve of Eratosthenes to produce primes less than n.
::    I used a map because it had much faster performance than a list.
::    Any key in the map is a non-prime. The value 1 indicates "false."
::    I.e. it's not a prime.
++  prime-map
  |=  n=@ud
  ^-  (map @ud @ud)
  =/  prime-map  `(map @ud @ud)`(my ~[[0 1] [1 1]])
  ::  start sieving with 2
  ::
  =/  sieve  2
  |-
  ::  if sieve is too large to be a factor we're done
  ::
  ?:  (gte (mul sieve sieve) n)
    prime-map
  ::  if not too large but not prime, move on
  ::
  ?:  =((~(gut by prime-map) sieve 0) 1)
    $(sieve +(sieve))
  ::  sequence: explanation
  ::
  ::    If s is the sieve number, we start sieving multiples
  ::    of s at s^2 in sequence: s^2, s^2 + s, s^2 + 2s, ...
  ::    We start at s^2 because any number smaller than s^2
  ::    has prime factors less than s and would have been
  ::    eliminated earlier in the sieving process.
  ::
  =/  sequence  (mul sieve sieve)
  |-
  ::  done sieving with s once sequence is past n
  ::
  ?:  (gte sequence n)
    ^$(sieve +(sieve))
  ::  if sequence position is known not prime we move on
  ::
  ?:  =((~(gut by prime-map) sequence 0) 1)
    $(sequence (add sequence sieve))
  ::  otherwise we mark position of sequence as not prime and move on
  ::
  $(prime-map (~(put by prime-map) sequence 1), sequence (add sequence sieve))
--

Java[edit]

public class RhondaNumbers {
    public static void main(String[] args) {
        final int limit = 15;
        for (int base = 2; base <= 36; ++base) {
            if (isPrime(base))
                continue;
            System.out.printf("First %d Rhonda numbers to base %d:\n", limit, base);
            int numbers[] = new int[limit];
            for (int n = 1, count = 0; count < limit; ++n) {
                if (isRhonda(base, n))
                    numbers[count++] = n;
            }
            System.out.printf("In base 10:");
            for (int i = 0; i < limit; ++i)
                System.out.printf(" %d", numbers[i]);
            System.out.printf("\nIn base %d:", base);
            for (int i = 0; i < limit; ++i)
                System.out.printf(" %s", Integer.toString(numbers[i], base));
            System.out.printf("\n\n");
        }
    }
    
    private static int digitProduct(int base, int n) {
        int product = 1;
        for (; n != 0; n /= base)
            product *= n % base;
        return product;
    }
     
    private static int primeFactorSum(int n) {
        int sum = 0;
        for (; (n & 1) == 0; n >>= 1)
            sum += 2;
        for (int p = 3; p * p <= n; p += 2)
            for (; n % p == 0; n /= p)
                sum += p;
        if (n > 1)
            sum += n;
        return sum;
    }
     
    private static boolean isPrime(int n) {
        if (n < 2)
            return false;
        if (n % 2 == 0)
            return n == 2;
        if (n % 3 == 0)
            return n == 3;
        for (int p = 5; p * p <= n; p += 4) {
            if (n % p == 0)
                return false;
            p += 2;
            if (n % p == 0)
                return false;
        }
        return true;
    }
     
    private static boolean isRhonda(int base, int n) {
        return digitProduct(base, n) == base * primeFactorSum(n);
    }
}
Output:
First 15 Rhonda numbers to base 4:
In base 10: 10206 11935 12150 16031 45030 94185 113022 114415 191149 244713 259753 374782 392121 503773 649902
In base 4: 2133132 2322133 2331312 3322133 22333212 112333221 123211332 123323233 232222231 323233221 333122221 1123133332 1133232321 1322333131 2132222232

First 15 Rhonda numbers to base 6:
In base 10: 855 1029 3813 5577 7040 7304 15104 19136 35350 36992 41031 42009 60368 65536 67821
In base 6: 3543 4433 25353 41453 52332 53452 153532 224332 431354 443132 513543 522253 1143252 1223224 1241553

First 15 Rhonda numbers to base 8:
In base 10: 1836 6318 6622 10530 14500 14739 17655 18550 25398 25956 30562 39215 39325 50875 51429
In base 8: 3454 14256 14736 24442 34244 34623 42367 44166 61466 62544 73542 114457 114635 143273 144345

First 15 Rhonda numbers to base 9:
In base 10: 15540 21054 25331 44360 44660 44733 47652 50560 54944 76857 77142 83334 83694 96448 97944
In base 9: 23276 31783 37665 66758 67232 67323 72326 76317 83328 126376 126733 136273 136723 156264 158316

First 15 Rhonda numbers to base 10:
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662

First 15 Rhonda numbers to base 12:
In base 10: 560 800 3993 4425 4602 4888 7315 8296 9315 11849 12028 13034 14828 15052 16264
In base 12: 3a8 568 2389 2689 27b6 29b4 4297 4974 5483 6a35 6b64 7662 86b8 8864 94b4

First 15 Rhonda numbers to base 14:
In base 10: 11475 18655 20565 29631 31725 45387 58404 58667 59950 63945 67525 68904 91245 99603 125543
In base 14: 4279 6b27 76cd ab27 b7c1 1277d 173da 17547 17bc2 19437 1a873 1b17a 25377 28427 33a75

First 15 Rhonda numbers to base 15:
In base 10: 2392 2472 11468 15873 17424 18126 19152 20079 24388 30758 31150 33004 33550 37925 39483
In base 15: a97 aec 35e8 4a83 5269 5586 5a1c 5e39 735d 91a8 936a 9ba4 9e1a b385 ba73

First 15 Rhonda numbers to base 16:
In base 10: 1000 1134 6776 15912 19624 20043 20355 23946 26296 29070 31906 32292 34236 34521 36465
In base 16: 3e8 46e 1a78 3e28 4ca8 4e4b 4f83 5d8a 66b8 718e 7ca2 7e24 85bc 86d9 8e71

First 15 Rhonda numbers to base 18:
In base 10: 1470 3000 8918 17025 19402 20650 21120 22156 26522 36549 38354 43281 46035 48768 54229
In base 18: 49c 94c 1998 2g9f 35fg 39d4 3b36 3e6g 49f8 64e9 6a6e 77a9 7g19 8696 956d

First 15 Rhonda numbers to base 20:
In base 10: 1815 11050 15295 21165 22165 30702 34510 34645 42292 44165 52059 53416 65945 78430 80712
In base 20: 4af 17ca 1i4f 2ci5 2f85 3gf2 465a 46c5 55ec 5a85 6a2j 6dag 84h5 9g1a a1fc

First 15 Rhonda numbers to base 21:
In base 10: 1632 5390 8512 12992 15678 25038 29412 34017 39552 48895 49147 61376 85078 89590 91798
In base 21: 3ef c4e j67 189e 1ebc 2eg6 33ec 3e2i 45e9 55i7 5697 6d3e 93j7 9e34 9j37

First 15 Rhonda numbers to base 22:
In base 10: 2695 4128 7865 28800 31710 37030 71875 74306 117760 117895 121626 126002 131427 175065 192753
In base 22: 5cb 8be g5b 2fb2 2lb8 3ab4 6gb1 6lbc b16g b1cj b96a bi78 c7bl g9fb i25b

First 15 Rhonda numbers to base 24:
In base 10: 2080 2709 3976 5628 5656 7144 8296 9030 10094 17612 20559 24616 26224 29106 31458
In base 24: 3eg 4gl 6lg 9ic 9jg c9g e9g fg6 hce 16dk 1bgf 1ihg 1lcg 22ci 26ei

First 15 Rhonda numbers to base 25:
In base 10: 6764 9633 13260 22022 53382 57640 66015 69006 97014 140130 142880 144235 159724 162565 165504
In base 25: ake fa8 l5a 1a5m 3aa7 3h5f 45ff 4aa6 655e 8o55 93f5 95ja a5do aa2f aek4

First 15 Rhonda numbers to base 26:
In base 10: 7788 9322 9374 11160 22165 27885 34905 44785 47385 49257 62517 72709 74217 108745 132302
In base 26: bde dke dme gd6 16kd 1f6d 1pgd 2e6d 2i2d 2kmd 3ecd 43ed 45kd 64md 7die

First 15 Rhonda numbers to base 27:
In base 10: 4797 11844 12078 13200 14841 17750 24320 26883 27477 46455 52750 58581 61009 61446 61500
In base 27: 6fi g6i gf9 i2o k9i o9b 169k 19ni 1aii 29jf 2i9j 2q9i 32ig 337l 339l

First 15 Rhonda numbers to base 28:
In base 10: 3094 5808 5832 7462 11160 13671 27270 28194 28638 39375 39550 49500 50862 52338 52938
In base 28: 3qe 7bc 7c8 9ee e6g hc7 16lq 17qq 18em 1m67 1mce 273o 28oe 2al6 2bei

First 15 Rhonda numbers to base 30:
In base 10: 3024 3168 5115 5346 5950 6762 7750 7956 8470 9476 9576 9849 10360 11495 13035
In base 30: 3ao 3fi 5kf 5s6 6ia 7fc 8ia 8p6 9ca afq aj6 as9 bfa cn5 eef

First 15 Rhonda numbers to base 32:
In base 10: 1944 3600 13520 15876 16732 16849 25410 25752 28951 47472 49610 50968 61596 64904 74005
In base 32: 1so 3gg d6g fg4 gas geh oq2 p4o s8n 1ebg 1gea 1hoo 1s4s 1vc8 288l

First 15 Rhonda numbers to base 33:
In base 10: 756 7040 7568 13826 24930 30613 59345 63555 64372 131427 227840 264044 313709 336385 344858
In base 33: mu 6fb 6vb cmw mtf s3m 1lgb 1pbu 1q3m 3lml 6b78 7bfb 8o2b 9btg 9jm8

First 15 Rhonda numbers to base 34:
In base 10: 5661 14161 15620 16473 22185 37145 125579 134692 135405 138472 140369 177086 250665 255552 295614
In base 34: 4uh c8h dhe e8h j6h w4h 36lh 3ehi 3f4h 3hqo 3jeh 4h6e 6csh 6h28 7hoi

First 15 Rhonda numbers to base 35:
In base 10: 8232 9476 9633 18634 30954 41905 52215 52440 56889 61992 62146 66339 98260 102180 103305
In base 35: 6p7 7pq 7u8 f7e p9e y7a 17lu 17sa 1bfe 1fl7 1fpl 1j5e 2a7f 2def 2ebk

First 15 Rhonda numbers to base 36:
In base 10: 1000 4800 5670 8190 10998 12412 13300 15750 16821 23016 51612 52734 67744 70929 75030
In base 36: rs 3pc 4di 6bi 8hi 9ks a9g c5i cz9 hrc 13to 14ou 1g9s 1iq9 1lw6

Julia[edit]

using Primes

isRhonda(n, b) = prod(digits(n, base=b)) == b * sum([prod(pair) for pair in factor(n).pe])

function displayrhondas(low, high, nshow)
    for b in filter(!isprime, low:high)
        n, rhondas = 1, Int[]
        while length(rhondas) < nshow
            isRhonda(n, b) && push!(rhondas, n)
            n += 1
        end
        println("First $nshow Rhondas in base $b:")
        println("In base 10: ", rhondas)
        println("In base $b: ", replace(string([string(i, base=b) for i in rhondas]), "\"" => ""), "\n")
    end
end

displayrhondas(2, 16, 15)
Output:
First 15 Rhondas in base 4:
In base 10: [10206, 11935, 12150, 16031, 45030, 94185, 113022, 114415, 191149, 244713, 259753, 374782, 392121, 503773, 649902]
In base 4: [2133132, 2322133, 2331312, 3322133, 22333212, 112333221, 123211332, 123323233, 232222231, 323233221, 333122221, 1123133332, 1133232321, 1322333131, 2132222232]

First 15 Rhondas in base 6:
In base 10: [855, 1029, 3813, 5577, 7040, 7304, 15104, 19136, 35350, 36992, 41031, 42009, 60368, 65536, 67821]
In base 6: [3543, 4433, 25353, 41453, 52332, 53452, 153532, 224332, 431354, 443132, 513543, 522253, 1143252, 1223224, 1241553]

First 15 Rhondas in base 8:
In base 10: [1836, 6318, 6622, 10530, 14500, 14739, 17655, 18550, 25398, 25956, 30562, 39215, 39325, 50875, 51429]
In base 8: [3454, 14256, 14736, 24442, 34244, 34623, 42367, 44166, 61466, 62544, 73542, 114457, 114635, 143273, 144345]

First 15 Rhondas in base 9:
In base 10: [15540, 21054, 25331, 44360, 44660, 44733, 47652, 50560, 54944, 76857, 77142, 83334, 83694, 96448, 97944]
In base 9: [23276, 31783, 37665, 66758, 67232, 67323, 72326, 76317, 83328, 126376, 126733, 136273, 136723, 156264, 158316]

First 15 Rhondas in base 10:
In base 10: [1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, 15625, 15698, 19435, 25284, 25662]
In base 10: [1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, 15625, 15698, 19435, 25284, 25662]

First 15 Rhondas in base 12:
In base 10: [560, 800, 3993, 4425, 4602, 4888, 7315, 8296, 9315, 11849, 12028, 13034, 14828, 15052, 16264]
In base 12: [3a8, 568, 2389, 2689, 27b6, 29b4, 4297, 4974, 5483, 6a35, 6b64, 7662, 86b8, 8864, 94b4]

First 15 Rhondas in base 14:
In base 10: [11475, 18655, 20565, 29631, 31725, 45387, 58404, 58667, 59950, 63945, 67525, 68904, 91245, 99603, 125543]
In base 14: [4279, 6b27, 76cd, ab27, b7c1, 1277d, 173da, 17547, 17bc2, 19437, 1a873, 1b17a, 25377, 28427, 33a75]

First 15 Rhondas in base 15:
In base 10: [2392, 2472, 11468, 15873, 17424, 18126, 19152, 20079, 24388, 30758, 31150, 33004, 33550, 37925, 39483]
In base 15: [a97, aec, 35e8, 4a83, 5269, 5586, 5a1c, 5e39, 735d, 91a8, 936a, 9ba4, 9e1a, b385, ba73]

First 15 Rhondas in base 16:
In base 10: [1000, 1134, 6776, 15912, 19624, 20043, 20355, 23946, 26296, 29070, 31906, 32292, 34236, 34521, 36465]
In base 16: [3e8, 46e, 1a78, 3e28, 4ca8, 4e4b, 4f83, 5d8a, 66b8, 718e, 7ca2, 7e24, 85bc, 86d9, 8e71]

Mathematica/Wolfram Language[edit]

ClearAll[RhondaNumberQ]
RhondaNumberQ[b_Integer][n_Integer] := Module[{l, r},
  l = Times @@ IntegerDigits[n, b];
  r = Total[Catenate[ConstantArray @@@ FactorInteger[n]]];
  l == b r
]
bases = Select[Range[2, 36], PrimeQ/*Not];
Do[
 Print["base ", b, ":", Take[Select[Range[700000], RhondaNumberQ[b]], UpTo[15]]];
 ,
 {b, bases}
]
Output:
base 4:{10206,11935,12150,16031,45030,94185,113022,114415,191149,244713,259753,374782,392121,503773,649902}
base 6:{855,1029,3813,5577,7040,7304,15104,19136,35350,36992,41031,42009,60368,65536,67821}
base 8:{1836,6318,6622,10530,14500,14739,17655,18550,25398,25956,30562,39215,39325,50875,51429}
base 9:{15540,21054,25331,44360,44660,44733,47652,50560,54944,76857,77142,83334,83694,96448,97944}
base 10:{1568,2835,4752,5265,5439,5664,5824,5832,8526,12985,15625,15698,19435,25284,25662}
base 12:{560,800,3993,4425,4602,4888,7315,8296,9315,11849,12028,13034,14828,15052,16264}
base 14:{11475,18655,20565,29631,31725,45387,58404,58667,59950,63945,67525,68904,91245,99603,125543}
base 15:{2392,2472,11468,15873,17424,18126,19152,20079,24388,30758,31150,33004,33550,37925,39483}
base 16:{1000,1134,6776,15912,19624,20043,20355,23946,26296,29070,31906,32292,34236,34521,36465}
base 18:{1470,3000,8918,17025,19402,20650,21120,22156,26522,36549,38354,43281,46035,48768,54229}
base 20:{1815,11050,15295,21165,22165,30702,34510,34645,42292,44165,52059,53416,65945,78430,80712}
base 21:{1632,5390,8512,12992,15678,25038,29412,34017,39552,48895,49147,61376,85078,89590,91798}
base 22:{2695,4128,7865,28800,31710,37030,71875,74306,117760,117895,121626,126002,131427,175065,192753}
base 24:{2080,2709,3976,5628,5656,7144,8296,9030,10094,17612,20559,24616,26224,29106,31458}
base 25:{6764,9633,13260,22022,53382,57640,66015,69006,97014,140130,142880,144235,159724,162565,165504}
base 26:{7788,9322,9374,11160,22165,27885,34905,44785,47385,49257,62517,72709,74217,108745,132302}
base 27:{4797,11844,12078,13200,14841,17750,24320,26883,27477,46455,52750,58581,61009,61446,61500}
base 28:{3094,5808,5832,7462,11160,13671,27270,28194,28638,39375,39550,49500,50862,52338,52938}
base 30:{3024,3168,5115,5346,5950,6762,7750,7956,8470,9476,9576,9849,10360,11495,13035}
base 32:{1944,3600,13520,15876,16732,16849,25410,25752,28951,47472,49610,50968,61596,64904,74005}
base 33:{756,7040,7568,13826,24930,30613,59345,63555,64372,131427,227840,264044,313709,336385,344858}
base 34:{5661,14161,15620,16473,22185,37145,125579,134692,135405,138472,140369,177086,250665,255552,295614}
base 35:{8232,9476,9633,18634,30954,41905,52215,52440,56889,61992,62146,66339,98260,102180,103305}
base 36:{1000,4800,5670,8190,10998,12412,13300,15750,16821,23016,51612,52734,67744,70929,75030}

Perl[edit]

Library: ntheory
use strict;
use warnings;
use feature 'say';
use ntheory qw<is_prime factor vecsum vecprod todigitstring todigits>;

sub rhonda {
    my($b, $cnt) = @_;
    my(@r,$n);
    while (++$n) {
        push @r, $n if ($b * vecsum factor($n)) == vecprod todigits($n,$b);
        return @r if $cnt == @r;
    }
}

for my $b (grep { ! is_prime $_ } 2..36) {
    my @Rb = map { todigitstring($_,$b) } my @R = rhonda($b, 15);
    say <<~EOT;
        First 15 Rhonda numbers to base $b:
        In base $b: @Rb
        In base 10: @R
        EOT
}
Output:
First 15 Rhonda numbers to base 4:
In base 4: 2133132 2322133 2331312 3322133 22333212 112333221 123211332 123323233 232222231 323233221 333122221 1123133332 1133232321 1322333131 2132222232
In base 10: 10206 11935 12150 16031 45030 94185 113022 114415 191149 244713 259753 374782 392121 503773 649902

First 15 Rhonda numbers to base 6:
In base 6: 3543 4433 25353 41453 52332 53452 153532 224332 431354 443132 513543 522253 1143252 1223224 1241553
In base 10: 855 1029 3813 5577 7040 7304 15104 19136 35350 36992 41031 42009 60368 65536 67821

First 15 Rhonda numbers to base 8:
In base 8: 3454 14256 14736 24442 34244 34623 42367 44166 61466 62544 73542 114457 114635 143273 144345
In base 10: 1836 6318 6622 10530 14500 14739 17655 18550 25398 25956 30562 39215 39325 50875 51429

First 15 Rhonda numbers to base 9:
In base 9: 23276 31783 37665 66758 67232 67323 72326 76317 83328 126376 126733 136273 136723 156264 158316
In base 10: 15540 21054 25331 44360 44660 44733 47652 50560 54944 76857 77142 83334 83694 96448 97944

First 15 Rhonda numbers to base 10:
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662

First 15 Rhonda numbers to base 12:
In base 12: 3a8 568 2389 2689 27b6 29b4 4297 4974 5483 6a35 6b64 7662 86b8 8864 94b4
In base 10: 560 800 3993 4425 4602 4888 7315 8296 9315 11849 12028 13034 14828 15052 16264

First 15 Rhonda numbers to base 14:
In base 14: 4279 6b27 76cd ab27 b7c1 1277d 173da 17547 17bc2 19437 1a873 1b17a 25377 28427 33a75
In base 10: 11475 18655 20565 29631 31725 45387 58404 58667 59950 63945 67525 68904 91245 99603 125543

First 15 Rhonda numbers to base 15:
In base 15: a97 aec 35e8 4a83 5269 5586 5a1c 5e39 735d 91a8 936a 9ba4 9e1a b385 ba73
In base 10: 2392 2472 11468 15873 17424 18126 19152 20079 24388 30758 31150 33004 33550 37925 39483

First 15 Rhonda numbers to base 16:
In base 16: 3e8 46e 1a78 3e28 4ca8 4e4b 4f83 5d8a 66b8 718e 7ca2 7e24 85bc 86d9 8e71
In base 10: 1000 1134 6776 15912 19624 20043 20355 23946 26296 29070 31906 32292 34236 34521 36465

First 15 Rhonda numbers to base 18:
In base 18: 49c 94c 1998 2g9f 35fg 39d4 3b36 3e6g 49f8 64e9 6a6e 77a9 7g19 8696 956d
In base 10: 1470 3000 8918 17025 19402 20650 21120 22156 26522 36549 38354 43281 46035 48768 54229

First 15 Rhonda numbers to base 20:
In base 20: 4af 17ca 1i4f 2ci5 2f85 3gf2 465a 46c5 55ec 5a85 6a2j 6dag 84h5 9g1a a1fc
In base 10: 1815 11050 15295 21165 22165 30702 34510 34645 42292 44165 52059 53416 65945 78430 80712

First 15 Rhonda numbers to base 21:
In base 21: 3ef c4e j67 189e 1ebc 2eg6 33ec 3e2i 45e9 55i7 5697 6d3e 93j7 9e34 9j37
In base 10: 1632 5390 8512 12992 15678 25038 29412 34017 39552 48895 49147 61376 85078 89590 91798

First 15 Rhonda numbers to base 22:
In base 22: 5cb 8be g5b 2fb2 2lb8 3ab4 6gb1 6lbc b16g b1cj b96a bi78 c7bl g9fb i25b
In base 10: 2695 4128 7865 28800 31710 37030 71875 74306 117760 117895 121626 126002 131427 175065 192753

First 15 Rhonda numbers to base 24:
In base 24: 3eg 4gl 6lg 9ic 9jg c9g e9g fg6 hce 16dk 1bgf 1ihg 1lcg 22ci 26ei
In base 10: 2080 2709 3976 5628 5656 7144 8296 9030 10094 17612 20559 24616 26224 29106 31458

First 15 Rhonda numbers to base 25:
In base 25: ake fa8 l5a 1a5m 3aa7 3h5f 45ff 4aa6 655e 8o55 93f5 95ja a5do aa2f aek4
In base 10: 6764 9633 13260 22022 53382 57640 66015 69006 97014 140130 142880 144235 159724 162565 165504

First 15 Rhonda numbers to base 26:
In base 26: bde dke dme gd6 16kd 1f6d 1pgd 2e6d 2i2d 2kmd 3ecd 43ed 45kd 64md 7die
In base 10: 7788 9322 9374 11160 22165 27885 34905 44785 47385 49257 62517 72709 74217 108745 132302

First 15 Rhonda numbers to base 27:
In base 27: 6fi g6i gf9 i2o k9i o9b 169k 19ni 1aii 29jf 2i9j 2q9i 32ig 337l 339l
In base 10: 4797 11844 12078 13200 14841 17750 24320 26883 27477 46455 52750 58581 61009 61446 61500

First 15 Rhonda numbers to base 28:
In base 28: 3qe 7bc 7c8 9ee e6g hc7 16lq 17qq 18em 1m67 1mce 273o 28oe 2al6 2bei
In base 10: 3094 5808 5832 7462 11160 13671 27270 28194 28638 39375 39550 49500 50862 52338 52938
First 15 Rhonda numbers to base 30:
In base 30: 3ao 3fi 5kf 5s6 6ia 7fc 8ia 8p6 9ca afq aj6 as9 bfa cn5 eef
In base 10: 3024 3168 5115 5346 5950 6762 7750 7956 8470 9476 9576 9849 10360 11495 13035

First 15 Rhonda numbers to base 32:
In base 32: 1so 3gg d6g fg4 gas geh oq2 p4o s8n 1ebg 1gea 1hoo 1s4s 1vc8 288l
In base 10: 1944 3600 13520 15876 16732 16849 25410 25752 28951 47472 49610 50968 61596 64904 74005

First 15 Rhonda numbers to base 33:
In base 33: mu 6fb 6vb cmw mtf s3m 1lgb 1pbu 1q3m 3lml 6b78 7bfb 8o2b 9btg 9jm8
In base 10: 756 7040 7568 13826 24930 30613 59345 63555 64372 131427 227840 264044 313709 336385 344858

First 15 Rhonda numbers to base 34:
In base 34: 4uh c8h dhe e8h j6h w4h 36lh 3ehi 3f4h 3hqo 3jeh 4h6e 6csh 6h28 7hoi
In base 10: 5661 14161 15620 16473 22185 37145 125579 134692 135405 138472 140369 177086 250665 255552 295614

First 15 Rhonda numbers to base 35:
In base 35: 6p7 7pq 7u8 f7e p9e y7a 17lu 17sa 1bfe 1fl7 1fpl 1j5e 2a7f 2def 2ebk
In base 10: 8232 9476 9633 18634 30954 41905 52215 52440 56889 61992 62146 66339 98260 102180 103305

First 15 Rhonda numbers to base 36:
In base 36: rs 3pc 4di 6bi 8hi 9ks a9g c5i cz9 hrc 13to 14ou 1g9s 1iq9 1lw6
In base 10: 1000 4800 5670 8190 10998 12412 13300 15750 16821 23016 51612 52734 67744 70929 75030

Phix[edit]

with javascript_semantics
constant fmt = """
First 15 Rhonda numbers in base %d:
In base 10:  %s
In base %-2d:  %s

"""
function digit(integer d) return d-iff(d<='9'?'0':'a'-10) end function

for base=2 to 36 do
    if not is_prime(base) then
        sequence rhondab = {},  -- (base)
                 rhondad = {}   -- (decimal)
        integer n = 1
        while length(rhondab)<15 do
            string digits = sprintf("%a",{{base,n}})
            if not find('0',digits)
            and (base!=10 or (find('5',digits) and sum(apply(digits,even))!=0)) then
                integer pd = product(apply(digits,digit)),
                        bs = base*sum(prime_factors(n,true,-1))
                if pd==bs then
                    string decdig = sprintf("%d",n)
                    integer l = max(length(decdig),length(digits))
                    rhondab = append(rhondab,pad_head(digits,l))
                    rhondad = append(rhondad,pad_head(decdig,l))
                end if
            end if
            n += 1
        end while
        printf(1,fmt,{base,join(rhondad),base,join(rhondab)})
    end if
end for
Output:
First 15 Rhonda numbers in base 4:
In base 10:    10206   11935   12150   16031    45030     94185    113022    114415    191149    244713    259753     374782     392121     503773     649902
In base 4 :  2133132 2322133 2331312 3322133 22333212 112333221 123211332 123323233 232222231 323233221 333122221 1123133332 1133232321 1322333131 2132222232

First 15 Rhonda numbers in base 6:
In base 10:   855 1029  3813  5577  7040  7304  15104  19136  35350  36992  41031  42009   60368   65536   67821
In base 6 :  3543 4433 25353 41453 52332 53452 153532 224332 431354 443132 513543 522253 1143252 1223224 1241553

First 15 Rhonda numbers in base 8:
In base 10:  1836  6318  6622 10530 14500 14739 17655 18550 25398 25956 30562  39215  39325  50875  51429
In base 8 :  3454 14256 14736 24442 34244 34623 42367 44166 61466 62544 73542 114457 114635 143273 144345

First 15 Rhonda numbers in base 9:
In base 10:  15540 21054 25331 44360 44660 44733 47652 50560 54944  76857  77142  83334  83694  96448  97944
In base 9 :  23276 31783 37665 66758 67232 67323 72326 76317 83328 126376 126733 136273 136723 156264 158316

First 15 Rhonda numbers in base 10:
In base 10:  1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662
In base 10:  1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662

First 15 Rhonda numbers in base 12:
In base 10:  560 800 3993 4425 4602 4888 7315 8296 9315 11849 12028 13034 14828 15052 16264
In base 12:  3a8 568 2389 2689 27b6 29b4 4297 4974 5483  6a35  6b64  7662  86b8  8864  94b4

First 15 Rhonda numbers in base 14:
In base 10:  11475 18655 20565 29631 31725 45387 58404 58667 59950 63945 67525 68904 91245 99603 125543
In base 14:   4279  6b27  76cd  ab27  b7c1 1277d 173da 17547 17bc2 19437 1a873 1b17a 25377 28427  33a75

First 15 Rhonda numbers in base 15:
In base 10:  2392 2472 11468 15873 17424 18126 19152 20079 24388 30758 31150 33004 33550 37925 39483
In base 15:   a97  aec  35e8  4a83  5269  5586  5a1c  5e39  735d  91a8  936a  9ba4  9e1a  b385  ba73

First 15 Rhonda numbers in base 16:
In base 10:  1000 1134 6776 15912 19624 20043 20355 23946 26296 29070 31906 32292 34236 34521 36465
In base 16:   3e8  46e 1a78  3e28  4ca8  4e4b  4f83  5d8a  66b8  718e  7ca2  7e24  85bc  86d9  8e71

First 15 Rhonda numbers in base 18:
In base 10:  1470 3000 8918 17025 19402 20650 21120 22156 26522 36549 38354 43281 46035 48768 54229
In base 18:   49c  94c 1998  2g9f  35fg  39d4  3b36  3e6g  49f8  64e9  6a6e  77a9  7g19  8696  956d

First 15 Rhonda numbers in base 20:
In base 10:  1815 11050 15295 21165 22165 30702 34510 34645 42292 44165 52059 53416 65945 78430 80712
In base 20:   4af  17ca  1i4f  2ci5  2f85  3gf2  465a  46c5  55ec  5a85  6a2j  6dag  84h5  9g1a  a1fc

First 15 Rhonda numbers in base 21:
In base 10:  1632 5390 8512 12992 15678 25038 29412 34017 39552 48895 49147 61376 85078 89590 91798
In base 21:   3ef  c4e  j67  189e  1ebc  2eg6  33ec  3e2i  45e9  55i7  5697  6d3e  93j7  9e34  9j37

First 15 Rhonda numbers in base 22:
In base 10:  2695 4128 7865 28800 31710 37030 71875 74306 117760 117895 121626 126002 131427 175065 192753
In base 22:   5cb  8be  g5b  2fb2  2lb8  3ab4  6gb1  6lbc   b16g   b1cj   b96a   bi78   c7bl   g9fb   i25b

First 15 Rhonda numbers in base 24:
In base 10:  2080 2709 3976 5628 5656 7144 8296 9030 10094 17612 20559 24616 26224 29106 31458
In base 24:   3eg  4gl  6lg  9ic  9jg  c9g  e9g  fg6   hce  16dk  1bgf  1ihg  1lcg  22ci  26ei

First 15 Rhonda numbers in base 25:
In base 10:  6764 9633 13260 22022 53382 57640 66015 69006 97014 140130 142880 144235 159724 162565 165504
In base 25:   ake  fa8   l5a  1a5m  3aa7  3h5f  45ff  4aa6  655e   8o55   93f5   95ja   a5do   aa2f   aek4

First 15 Rhonda numbers in base 26:
In base 10:  7788 9322 9374 11160 22165 27885 34905 44785 47385 49257 62517 72709 74217 108745 132302
In base 26:   bde  dke  dme   gd6  16kd  1f6d  1pgd  2e6d  2i2d  2kmd  3ecd  43ed  45kd   64md   7die

First 15 Rhonda numbers in base 27:
In base 10:  4797 11844 12078 13200 14841 17750 24320 26883 27477 46455 52750 58581 61009 61446 61500
In base 27:   6fi   g6i   gf9   i2o   k9i   o9b  169k  19ni  1aii  29jf  2i9j  2q9i  32ig  337l  339l

First 15 Rhonda numbers in base 28:
In base 10:  3094 5808 5832 7462 11160 13671 27270 28194 28638 39375 39550 49500 50862 52338 52938
In base 28:   3qe  7bc  7c8  9ee   e6g   hc7  16lq  17qq  18em  1m67  1mce  273o  28oe  2al6  2bei

First 15 Rhonda numbers in base 30:
In base 10:  3024 3168 5115 5346 5950 6762 7750 7956 8470 9476 9576 9849 10360 11495 13035
In base 30:   3ao  3fi  5kf  5s6  6ia  7fc  8ia  8p6  9ca  afq  aj6  as9   bfa   cn5   eef

First 15 Rhonda numbers in base 32:
In base 10:  1944 3600 13520 15876 16732 16849 25410 25752 28951 47472 49610 50968 61596 64904 74005
In base 32:   1so  3gg   d6g   fg4   gas   geh   oq2   p4o   s8n  1ebg  1gea  1hoo  1s4s  1vc8  288l

First 15 Rhonda numbers in base 33:
In base 10:  756 7040 7568 13826 24930 30613 59345 63555 64372 131427 227840 264044 313709 336385 344858
In base 33:   mu  6fb  6vb   cmw   mtf   s3m  1lgb  1pbu  1q3m   3lml   6b78   7bfb   8o2b   9btg   9jm8

First 15 Rhonda numbers in base 34:
In base 10:  5661 14161 15620 16473 22185 37145 125579 134692 135405 138472 140369 177086 250665 255552 295614
In base 34:   4uh   c8h   dhe   e8h   j6h   w4h   36lh   3ehi   3f4h   3hqo   3jeh   4h6e   6csh   6h28   7hoi

First 15 Rhonda numbers in base 35:
In base 10:  8232 9476 9633 18634 30954 41905 52215 52440 56889 61992 62146 66339 98260 102180 103305
In base 35:   6p7  7pq  7u8   f7e   p9e   y7a  17lu  17sa  1bfe  1fl7  1fpl  1j5e  2a7f   2def   2ebk

First 15 Rhonda numbers in base 36:
In base 10:  1000 4800 5670 8190 10998 12412 13300 15750 16821 23016 51612 52734 67744 70929 75030
In base 36:    rs  3pc  4di  6bi   8hi   9ks   a9g   c5i   cz9   hrc  13to  14ou  1g9s  1iq9  1lw6

Raku[edit]

Find and show the first 15 so as to display the namesake Rhonda number 25662.

use Prime::Factor;

my @factor-sum;

@factor-sum[1000000] = 42; # Sink a large index to make access thread safe 

sub rhonda ($base) {
    (1..∞).hyper.map: { $_ if $base * (@factor-sum[$_] //= .&prime-factors.sum) == [×] .polymod($base xx *) }
}

for (flat 2..16, 17..36).grep: { !.&is-prime }  -> $b {
    put "\nFirst 15 Rhonda numbers to base $b:";
    my @rhonda = rhonda($b)[^15];
    my $ch = @rhonda[*-1].chars max @rhonda[*-1].base($b).chars;
    put "In base 10: " ~ @rhonda».fmt("%{$ch}s").join: ', ';
    put $b.fmt("In base %2d: ") ~ @rhonda».base($b)».fmt("%{$ch}s").join: ', ';
}
Output:
First 15 Rhonda numbers to base 4:
In base 10:      10206,      11935,      12150,      16031,      45030,      94185,     113022,     114415,     191149,     244713,     259753,     374782,     392121,     503773,     649902
In base  4:    2133132,    2322133,    2331312,    3322133,   22333212,  112333221,  123211332,  123323233,  232222231,  323233221,  333122221, 1123133332, 1133232321, 1322333131, 2132222232

First 15 Rhonda numbers to base 6:
In base 10:     855,    1029,    3813,    5577,    7040,    7304,   15104,   19136,   35350,   36992,   41031,   42009,   60368,   65536,   67821
In base  6:    3543,    4433,   25353,   41453,   52332,   53452,  153532,  224332,  431354,  443132,  513543,  522253, 1143252, 1223224, 1241553

First 15 Rhonda numbers to base 8:
In base 10:   1836,   6318,   6622,  10530,  14500,  14739,  17655,  18550,  25398,  25956,  30562,  39215,  39325,  50875,  51429
In base  8:   3454,  14256,  14736,  24442,  34244,  34623,  42367,  44166,  61466,  62544,  73542, 114457, 114635, 143273, 144345

First 15 Rhonda numbers to base 9:
In base 10:  15540,  21054,  25331,  44360,  44660,  44733,  47652,  50560,  54944,  76857,  77142,  83334,  83694,  96448,  97944
In base  9:  23276,  31783,  37665,  66758,  67232,  67323,  72326,  76317,  83328, 126376, 126733, 136273, 136723, 156264, 158316

First 15 Rhonda numbers to base 10:
In base 10:  1568,  2835,  4752,  5265,  5439,  5664,  5824,  5832,  8526, 12985, 15625, 15698, 19435, 25284, 25662
In base 10:  1568,  2835,  4752,  5265,  5439,  5664,  5824,  5832,  8526, 12985, 15625, 15698, 19435, 25284, 25662

First 15 Rhonda numbers to base 12:
In base 10:   560,   800,  3993,  4425,  4602,  4888,  7315,  8296,  9315, 11849, 12028, 13034, 14828, 15052, 16264
In base 12:   3A8,   568,  2389,  2689,  27B6,  29B4,  4297,  4974,  5483,  6A35,  6B64,  7662,  86B8,  8864,  94B4

First 15 Rhonda numbers to base 14:
In base 10:  11475,  18655,  20565,  29631,  31725,  45387,  58404,  58667,  59950,  63945,  67525,  68904,  91245,  99603, 125543
In base 14:   4279,   6B27,   76CD,   AB27,   B7C1,  1277D,  173DA,  17547,  17BC2,  19437,  1A873,  1B17A,  25377,  28427,  33A75

First 15 Rhonda numbers to base 15:
In base 10:  2392,  2472, 11468, 15873, 17424, 18126, 19152, 20079, 24388, 30758, 31150, 33004, 33550, 37925, 39483
In base 15:   A97,   AEC,  35E8,  4A83,  5269,  5586,  5A1C,  5E39,  735D,  91A8,  936A,  9BA4,  9E1A,  B385,  BA73

First 15 Rhonda numbers to base 16:
In base 10:  1000,  1134,  6776, 15912, 19624, 20043, 20355, 23946, 26296, 29070, 31906, 32292, 34236, 34521, 36465
In base 16:   3E8,   46E,  1A78,  3E28,  4CA8,  4E4B,  4F83,  5D8A,  66B8,  718E,  7CA2,  7E24,  85BC,  86D9,  8E71

First 15 Rhonda numbers to base 18:
In base 10:  1470,  3000,  8918, 17025, 19402, 20650, 21120, 22156, 26522, 36549, 38354, 43281, 46035, 48768, 54229
In base 18:   49C,   94C,  1998,  2G9F,  35FG,  39D4,  3B36,  3E6G,  49F8,  64E9,  6A6E,  77A9,  7G19,  8696,  956D

First 15 Rhonda numbers to base 20:
In base 10:  1815, 11050, 15295, 21165, 22165, 30702, 34510, 34645, 42292, 44165, 52059, 53416, 65945, 78430, 80712
In base 20:   4AF,  17CA,  1I4F,  2CI5,  2F85,  3GF2,  465A,  46C5,  55EC,  5A85,  6A2J,  6DAG,  84H5,  9G1A,  A1FC

First 15 Rhonda numbers to base 21:
In base 10:  1632,  5390,  8512, 12992, 15678, 25038, 29412, 34017, 39552, 48895, 49147, 61376, 85078, 89590, 91798
In base 21:   3EF,   C4E,   J67,  189E,  1EBC,  2EG6,  33EC,  3E2I,  45E9,  55I7,  5697,  6D3E,  93J7,  9E34,  9J37

First 15 Rhonda numbers to base 22:
In base 10:   2695,   4128,   7865,  28800,  31710,  37030,  71875,  74306, 117760, 117895, 121626, 126002, 131427, 175065, 192753
In base 22:    5CB,    8BE,    G5B,   2FB2,   2LB8,   3AB4,   6GB1,   6LBC,   B16G,   B1CJ,   B96A,   BI78,   C7BL,   G9FB,   I25B

First 15 Rhonda numbers to base 24:
In base 10:  2080,  2709,  3976,  5628,  5656,  7144,  8296,  9030, 10094, 17612, 20559, 24616, 26224, 29106, 31458
In base 24:   3EG,   4GL,   6LG,   9IC,   9JG,   C9G,   E9G,   FG6,   HCE,  16DK,  1BGF,  1IHG,  1LCG,  22CI,  26EI

First 15 Rhonda numbers to base 25:
In base 10:   6764,   9633,  13260,  22022,  53382,  57640,  66015,  69006,  97014, 140130, 142880, 144235, 159724, 162565, 165504
In base 25:    AKE,    FA8,    L5A,   1A5M,   3AA7,   3H5F,   45FF,   4AA6,   655E,   8O55,   93F5,   95JA,   A5DO,   AA2F,   AEK4

First 15 Rhonda numbers to base 26:
In base 10:   7788,   9322,   9374,  11160,  22165,  27885,  34905,  44785,  47385,  49257,  62517,  72709,  74217, 108745, 132302
In base 26:    BDE,    DKE,    DME,    GD6,   16KD,   1F6D,   1PGD,   2E6D,   2I2D,   2KMD,   3ECD,   43ED,   45KD,   64MD,   7DIE

First 15 Rhonda numbers to base 27:
In base 10:  4797, 11844, 12078, 13200, 14841, 17750, 24320, 26883, 27477, 46455, 52750, 58581, 61009, 61446, 61500
In base 27:   6FI,   G6I,   GF9,   I2O,   K9I,   O9B,  169K,  19NI,  1AII,  29JF,  2I9J,  2Q9I,  32IG,  337L,  339L

First 15 Rhonda numbers to base 28:
In base 10:  3094,  5808,  5832,  7462, 11160, 13671, 27270, 28194, 28638, 39375, 39550, 49500, 50862, 52338, 52938
In base 28:   3QE,   7BC,   7C8,   9EE,   E6G,   HC7,  16LQ,  17QQ,  18EM,  1M67,  1MCE,  273O,  28OE,  2AL6,  2BEI

First 15 Rhonda numbers to base 30:
In base 10:  3024,  3168,  5115,  5346,  5950,  6762,  7750,  7956,  8470,  9476,  9576,  9849, 10360, 11495, 13035
In base 30:   3AO,   3FI,   5KF,   5S6,   6IA,   7FC,   8IA,   8P6,   9CA,   AFQ,   AJ6,   AS9,   BFA,   CN5,   EEF

First 15 Rhonda numbers to base 32:
In base 10:  1944,  3600, 13520, 15876, 16732, 16849, 25410, 25752, 28951, 47472, 49610, 50968, 61596, 64904, 74005
In base 32:   1SO,   3GG,   D6G,   FG4,   GAS,   GEH,   OQ2,   P4O,   S8N,  1EBG,  1GEA,  1HOO,  1S4S,  1VC8,  288L

First 15 Rhonda numbers to base 33:
In base 10:    756,   7040,   7568,  13826,  24930,  30613,  59345,  63555,  64372, 131427, 227840, 264044, 313709, 336385, 344858
In base 33:     MU,    6FB,    6VB,    CMW,    MTF,    S3M,   1LGB,   1PBU,   1Q3M,   3LML,   6B78,   7BFB,   8O2B,   9BTG,   9JM8

First 15 Rhonda numbers to base 34:
In base 10:   5661,  14161,  15620,  16473,  22185,  37145, 125579, 134692, 135405, 138472, 140369, 177086, 250665, 255552, 295614
In base 34:    4UH,    C8H,    DHE,    E8H,    J6H,    W4H,   36LH,   3EHI,   3F4H,   3HQO,   3JEH,   4H6E,   6CSH,   6H28,   7HOI

First 15 Rhonda numbers to base 35:
In base 10:   8232,   9476,   9633,  18634,  30954,  41905,  52215,  52440,  56889,  61992,  62146,  66339,  98260, 102180, 103305
In base 35:    6P7,    7PQ,    7U8,    F7E,    P9E,    Y7A,   17LU,   17SA,   1BFE,   1FL7,   1FPL,   1J5E,   2A7F,   2DEF,   2EBK

First 15 Rhonda numbers to base 36:
In base 10:  1000,  4800,  5670,  8190, 10998, 12412, 13300, 15750, 16821, 23016, 51612, 52734, 67744, 70929, 75030
In base 36:    RS,   3PC,   4DI,   6BI,   8HI,   9KS,   A9G,   C5I,   CZ9,   HRC,  13TO,  14OU,  1G9S,  1IQ9,  1LW6

Rust[edit]

// [dependencies]
// radix_fmt = "1.0"

fn digit_product(base: u32, mut n: u32) -> u32 {
    let mut product = 1;
    while n != 0 {
        product *= n % base;
        n /= base;
    }
    product
}

fn prime_factor_sum(mut n: u32) -> u32 {
    let mut sum = 0;
    while (n & 1) == 0 {
        sum += 2;
        n >>= 1;
    }
    let mut p = 3;
    while p * p <= n {
        while n % p == 0 {
            sum += p;
            n /= p;
        }
        p += 2;
    }
    if n > 1 {
        sum += n;
    }
    sum
}

fn is_prime(n: u32) -> bool {
    if n < 2 {
        return false;
    }
    if n % 2 == 0 {
        return n == 2;
    }
    if n % 3 == 0 {
        return n == 3;
    }
    let mut p = 5;
    while p * p <= n {
        if n % p == 0 {
            return false;
        }
        p += 2;
        if n % p == 0 {
            return false;
        }
        p += 4;
    }
    true
}

fn is_rhonda(base: u32, n: u32) -> bool {
    digit_product(base, n) == base * prime_factor_sum(n)
}

fn main() {
    let limit = 15;
    for base in 2..=36 {
        if is_prime(base) {
            continue;
        }
        println!("First {} Rhonda numbers to base {}:", limit, base);
        let numbers: Vec<u32> = (1..).filter(|x| is_rhonda(base, *x)).take(limit).collect();
        print!("In base 10:");
        for n in &numbers {
            print!(" {}", n);
        }
        print!("\nIn base {}:", base);
        for n in &numbers {
            print!(" {}", radix_fmt::radix(*n, base as u8));
        }
        print!("\n\n");
    }
}
Output:
First 15 Rhonda numbers to base 4:
In base 10: 10206 11935 12150 16031 45030 94185 113022 114415 191149 244713 259753 374782 392121 503773 649902
In base 4: 2133132 2322133 2331312 3322133 22333212 112333221 123211332 123323233 232222231 323233221 333122221 1123133332 1133232321 1322333131 2132222232

First 15 Rhonda numbers to base 6:
In base 10: 855 1029 3813 5577 7040 7304 15104 19136 35350 36992 41031 42009 60368 65536 67821
In base 6: 3543 4433 25353 41453 52332 53452 153532 224332 431354 443132 513543 522253 1143252 1223224 1241553

First 15 Rhonda numbers to base 8:
In base 10: 1836 6318 6622 10530 14500 14739 17655 18550 25398 25956 30562 39215 39325 50875 51429
In base 8: 3454 14256 14736 24442 34244 34623 42367 44166 61466 62544 73542 114457 114635 143273 144345

First 15 Rhonda numbers to base 9:
In base 10: 15540 21054 25331 44360 44660 44733 47652 50560 54944 76857 77142 83334 83694 96448 97944
In base 9: 23276 31783 37665 66758 67232 67323 72326 76317 83328 126376 126733 136273 136723 156264 158316

First 15 Rhonda numbers to base 10:
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662

First 15 Rhonda numbers to base 12:
In base 10: 560 800 3993 4425 4602 4888 7315 8296 9315 11849 12028 13034 14828 15052 16264
In base 12: 3a8 568 2389 2689 27b6 29b4 4297 4974 5483 6a35 6b64 7662 86b8 8864 94b4

First 15 Rhonda numbers to base 14:
In base 10: 11475 18655 20565 29631 31725 45387 58404 58667 59950 63945 67525 68904 91245 99603 125543
In base 14: 4279 6b27 76cd ab27 b7c1 1277d 173da 17547 17bc2 19437 1a873 1b17a 25377 28427 33a75

First 15 Rhonda numbers to base 15:
In base 10: 2392 2472 11468 15873 17424 18126 19152 20079 24388 30758 31150 33004 33550 37925 39483
In base 15: a97 aec 35e8 4a83 5269 5586 5a1c 5e39 735d 91a8 936a 9ba4 9e1a b385 ba73

First 15 Rhonda numbers to base 16:
In base 10: 1000 1134 6776 15912 19624 20043 20355 23946 26296 29070 31906 32292 34236 34521 36465
In base 16: 3e8 46e 1a78 3e28 4ca8 4e4b 4f83 5d8a 66b8 718e 7ca2 7e24 85bc 86d9 8e71

First 15 Rhonda numbers to base 18:
In base 10: 1470 3000 8918 17025 19402 20650 21120 22156 26522 36549 38354 43281 46035 48768 54229
In base 18: 49c 94c 1998 2g9f 35fg 39d4 3b36 3e6g 49f8 64e9 6a6e 77a9 7g19 8696 956d

First 15 Rhonda numbers to base 20:
In base 10: 1815 11050 15295 21165 22165 30702 34510 34645 42292 44165 52059 53416 65945 78430 80712
In base 20: 4af 17ca 1i4f 2ci5 2f85 3gf2 465a 46c5 55ec 5a85 6a2j 6dag 84h5 9g1a a1fc

First 15 Rhonda numbers to base 21:
In base 10: 1632 5390 8512 12992 15678 25038 29412 34017 39552 48895 49147 61376 85078 89590 91798
In base 21: 3ef c4e j67 189e 1ebc 2eg6 33ec 3e2i 45e9 55i7 5697 6d3e 93j7 9e34 9j37

First 15 Rhonda numbers to base 22:
In base 10: 2695 4128 7865 28800 31710 37030 71875 74306 117760 117895 121626 126002 131427 175065 192753
In base 22: 5cb 8be g5b 2fb2 2lb8 3ab4 6gb1 6lbc b16g b1cj b96a bi78 c7bl g9fb i25b

First 15 Rhonda numbers to base 24:
In base 10: 2080 2709 3976 5628 5656 7144 8296 9030 10094 17612 20559 24616 26224 29106 31458
In base 24: 3eg 4gl 6lg 9ic 9jg c9g e9g fg6 hce 16dk 1bgf 1ihg 1lcg 22ci 26ei

First 15 Rhonda numbers to base 25:
In base 10: 6764 9633 13260 22022 53382 57640 66015 69006 97014 140130 142880 144235 159724 162565 165504
In base 25: ake fa8 l5a 1a5m 3aa7 3h5f 45ff 4aa6 655e 8o55 93f5 95ja a5do aa2f aek4

First 15 Rhonda numbers to base 26:
In base 10: 7788 9322 9374 11160 22165 27885 34905 44785 47385 49257 62517 72709 74217 108745 132302
In base 26: bde dke dme gd6 16kd 1f6d 1pgd 2e6d 2i2d 2kmd 3ecd 43ed 45kd 64md 7die

First 15 Rhonda numbers to base 27:
In base 10: 4797 11844 12078 13200 14841 17750 24320 26883 27477 46455 52750 58581 61009 61446 61500
In base 27: 6fi g6i gf9 i2o k9i o9b 169k 19ni 1aii 29jf 2i9j 2q9i 32ig 337l 339l

First 15 Rhonda numbers to base 28:
In base 10: 3094 5808 5832 7462 11160 13671 27270 28194 28638 39375 39550 49500 50862 52338 52938
In base 28: 3qe 7bc 7c8 9ee e6g hc7 16lq 17qq 18em 1m67 1mce 273o 28oe 2al6 2bei

First 15 Rhonda numbers to base 30:
In base 10: 3024 3168 5115 5346 5950 6762 7750 7956 8470 9476 9576 9849 10360 11495 13035
In base 30: 3ao 3fi 5kf 5s6 6ia 7fc 8ia 8p6 9ca afq aj6 as9 bfa cn5 eef

First 15 Rhonda numbers to base 32:
In base 10: 1944 3600 13520 15876 16732 16849 25410 25752 28951 47472 49610 50968 61596 64904 74005
In base 32: 1so 3gg d6g fg4 gas geh oq2 p4o s8n 1ebg 1gea 1hoo 1s4s 1vc8 288l

First 15 Rhonda numbers to base 33:
In base 10: 756 7040 7568 13826 24930 30613 59345 63555 64372 131427 227840 264044 313709 336385 344858
In base 33: mu 6fb 6vb cmw mtf s3m 1lgb 1pbu 1q3m 3lml 6b78 7bfb 8o2b 9btg 9jm8

First 15 Rhonda numbers to base 34:
In base 10: 5661 14161 15620 16473 22185 37145 125579 134692 135405 138472 140369 177086 250665 255552 295614
In base 34: 4uh c8h dhe e8h j6h w4h 36lh 3ehi 3f4h 3hqo 3jeh 4h6e 6csh 6h28 7hoi

First 15 Rhonda numbers to base 35:
In base 10: 8232 9476 9633 18634 30954 41905 52215 52440 56889 61992 62146 66339 98260 102180 103305
In base 35: 6p7 7pq 7u8 f7e p9e y7a 17lu 17sa 1bfe 1fl7 1fpl 1j5e 2a7f 2def 2ebk

First 15 Rhonda numbers to base 36:
In base 10: 1000 4800 5670 8190 10998 12412 13300 15750 16821 23016 51612 52734 67744 70929 75030
In base 36: rs 3pc 4di 6bi 8hi 9ks a9g c5i cz9 hrc 13to 14ou 1g9s 1iq9 1lw6

Sidef[edit]

func is_rhonda_number(n, base = 10) {
    base.is_composite || return false
    n > 0             || return false
    n.digits(base).prod == base*n.factor.sum
}

for b in (2..16 -> grep { .is_composite }) {
    say ("First 10 Rhonda numbers to base #{b}: ",
        10.by { is_rhonda_number(_, b) })
}
Output:
First 10 Rhonda numbers to base 4: [10206, 11935, 12150, 16031, 45030, 94185, 113022, 114415, 191149, 244713]
First 10 Rhonda numbers to base 6: [855, 1029, 3813, 5577, 7040, 7304, 15104, 19136, 35350, 36992]
First 10 Rhonda numbers to base 8: [1836, 6318, 6622, 10530, 14500, 14739, 17655, 18550, 25398, 25956]
First 10 Rhonda numbers to base 9: [15540, 21054, 25331, 44360, 44660, 44733, 47652, 50560, 54944, 76857]
First 10 Rhonda numbers to base 10: [1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985]
First 10 Rhonda numbers to base 12: [560, 800, 3993, 4425, 4602, 4888, 7315, 8296, 9315, 11849]
First 10 Rhonda numbers to base 14: [11475, 18655, 20565, 29631, 31725, 45387, 58404, 58667, 59950, 63945]
First 10 Rhonda numbers to base 15: [2392, 2472, 11468, 15873, 17424, 18126, 19152, 20079, 24388, 30758]
First 10 Rhonda numbers to base 16: [1000, 1134, 6776, 15912, 19624, 20043, 20355, 23946, 26296, 29070]

Swift[edit]

func digitProduct(base: Int, num: Int) -> Int {
    var product = 1
    var n = num
    while n != 0 {
        product *= n % base
        n /= base
    }
    return product
}

func primeFactorSum(_ num: Int) -> Int {
    var sum = 0
    var n = num
    while (n & 1) == 0 {
        sum += 2
        n >>= 1
    }
    var p = 3
    while p * p <= n {
        while n % p == 0 {
            sum += p
            n /= p
        }
        p += 2
    }
    if n > 1 {
        sum += n
    }
    return sum
}

func isPrime(_ n: Int) -> Bool {
    if n < 2 {
        return false
    }
    if n % 2 == 0 {
        return n == 2
    }
    if n % 3 == 0 {
        return n == 3
    }
    var p = 5
    while p * p <= n {
        if n % p == 0 {
            return false
        }
        p += 2
        if n % p == 0 {
            return false
        }
        p += 4
    }
    return true
}

func isRhonda(base: Int, num: Int) -> Bool {
    return digitProduct(base: base, num: num) == base * primeFactorSum(num)
}

let limit = 15
for base in 2...36 {
    if isPrime(base) {
        continue
    }
    print("First \(limit) Rhonda numbers to base \(base):")
    let numbers = Array((1...).lazy.filter{ isRhonda(base: base, num: $0) }.prefix(limit))
    print("In base 10:", terminator: "")
    for n in numbers {
        print(" \(n)", terminator: "")
    }
    print("\nIn base \(base):", terminator: "")
    for n in numbers {
        print(" \(String(n, radix: base))", terminator: "")
    }
    print("\n")
}
Output:
First 15 Rhonda numbers to base 4:
In base 10: 10206 11935 12150 16031 45030 94185 113022 114415 191149 244713 259753 374782 392121 503773 649902
In base 4: 2133132 2322133 2331312 3322133 22333212 112333221 123211332 123323233 232222231 323233221 333122221 1123133332 1133232321 1322333131 2132222232

First 15 Rhonda numbers to base 6:
In base 10: 855 1029 3813 5577 7040 7304 15104 19136 35350 36992 41031 42009 60368 65536 67821
In base 6: 3543 4433 25353 41453 52332 53452 153532 224332 431354 443132 513543 522253 1143252 1223224 1241553

First 15 Rhonda numbers to base 8:
In base 10: 1836 6318 6622 10530 14500 14739 17655 18550 25398 25956 30562 39215 39325 50875 51429
In base 8: 3454 14256 14736 24442 34244 34623 42367 44166 61466 62544 73542 114457 114635 143273 144345

First 15 Rhonda numbers to base 9:
In base 10: 15540 21054 25331 44360 44660 44733 47652 50560 54944 76857 77142 83334 83694 96448 97944
In base 9: 23276 31783 37665 66758 67232 67323 72326 76317 83328 126376 126733 136273 136723 156264 158316

First 15 Rhonda numbers to base 10:
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662
In base 10: 1568 2835 4752 5265 5439 5664 5824 5832 8526 12985 15625 15698 19435 25284 25662

First 15 Rhonda numbers to base 12:
In base 10: 560 800 3993 4425 4602 4888 7315 8296 9315 11849 12028 13034 14828 15052 16264
In base 12: 3a8 568 2389 2689 27b6 29b4 4297 4974 5483 6a35 6b64 7662 86b8 8864 94b4

First 15 Rhonda numbers to base 14:
In base 10: 11475 18655 20565 29631 31725 45387 58404 58667 59950 63945 67525 68904 91245 99603 125543
In base 14: 4279 6b27 76cd ab27 b7c1 1277d 173da 17547 17bc2 19437 1a873 1b17a 25377 28427 33a75

First 15 Rhonda numbers to base 15:
In base 10: 2392 2472 11468 15873 17424 18126 19152 20079 24388 30758 31150 33004 33550 37925 39483
In base 15: a97 aec 35e8 4a83 5269 5586 5a1c 5e39 735d 91a8 936a 9ba4 9e1a b385 ba73

First 15 Rhonda numbers to base 16:
In base 10: 1000 1134 6776 15912 19624 20043 20355 23946 26296 29070 31906 32292 34236 34521 36465
In base 16: 3e8 46e 1a78 3e28 4ca8 4e4b 4f83 5d8a 66b8 718e 7ca2 7e24 85bc 86d9 8e71

First 15 Rhonda numbers to base 18:
In base 10: 1470 3000 8918 17025 19402 20650 21120 22156 26522 36549 38354 43281 46035 48768 54229
In base 18: 49c 94c 1998 2g9f 35fg 39d4 3b36 3e6g 49f8 64e9 6a6e 77a9 7g19 8696 956d

First 15 Rhonda numbers to base 20:
In base 10: 1815 11050 15295 21165 22165 30702 34510 34645 42292 44165 52059 53416 65945 78430 80712
In base 20: 4af 17ca 1i4f 2ci5 2f85 3gf2 465a 46c5 55ec 5a85 6a2j 6dag 84h5 9g1a a1fc

First 15 Rhonda numbers to base 21:
In base 10: 1632 5390 8512 12992 15678 25038 29412 34017 39552 48895 49147 61376 85078 89590 91798
In base 21: 3ef c4e j67 189e 1ebc 2eg6 33ec 3e2i 45e9 55i7 5697 6d3e 93j7 9e34 9j37

First 15 Rhonda numbers to base 22:
In base 10: 2695 4128 7865 28800 31710 37030 71875 74306 117760 117895 121626 126002 131427 175065 192753
In base 22: 5cb 8be g5b 2fb2 2lb8 3ab4 6gb1 6lbc b16g b1cj b96a bi78 c7bl g9fb i25b

First 15 Rhonda numbers to base 24:
In base 10: 2080 2709 3976 5628 5656 7144 8296 9030 10094 17612 20559 24616 26224 29106 31458
In base 24: 3eg 4gl 6lg 9ic 9jg c9g e9g fg6 hce 16dk 1bgf 1ihg 1lcg 22ci 26ei

First 15 Rhonda numbers to base 25:
In base 10: 6764 9633 13260 22022 53382 57640 66015 69006 97014 140130 142880 144235 159724 162565 165504
In base 25: ake fa8 l5a 1a5m 3aa7 3h5f 45ff 4aa6 655e 8o55 93f5 95ja a5do aa2f aek4

First 15 Rhonda numbers to base 26:
In base 10: 7788 9322 9374 11160 22165 27885 34905 44785 47385 49257 62517 72709 74217 108745 132302
In base 26: bde dke dme gd6 16kd 1f6d 1pgd 2e6d 2i2d 2kmd 3ecd 43ed 45kd 64md 7die

First 15 Rhonda numbers to base 27:
In base 10: 4797 11844 12078 13200 14841 17750 24320 26883 27477 46455 52750 58581 61009 61446 61500
In base 27: 6fi g6i gf9 i2o k9i o9b 169k 19ni 1aii 29jf 2i9j 2q9i 32ig 337l 339l

First 15 Rhonda numbers to base 28:
In base 10: 3094 5808 5832 7462 11160 13671 27270 28194 28638 39375 39550 49500 50862 52338 52938
In base 28: 3qe 7bc 7c8 9ee e6g hc7 16lq 17qq 18em 1m67 1mce 273o 28oe 2al6 2bei

First 15 Rhonda numbers to base 30:
In base 10: 3024 3168 5115 5346 5950 6762 7750 7956 8470 9476 9576 9849 10360 11495 13035
In base 30: 3ao 3fi 5kf 5s6 6ia 7fc 8ia 8p6 9ca afq aj6 as9 bfa cn5 eef

First 15 Rhonda numbers to base 32:
In base 10: 1944 3600 13520 15876 16732 16849 25410 25752 28951 47472 49610 50968 61596 64904 74005
In base 32: 1so 3gg d6g fg4 gas geh oq2 p4o s8n 1ebg 1gea 1hoo 1s4s 1vc8 288l

First 15 Rhonda numbers to base 33:
In base 10: 756 7040 7568 13826 24930 30613 59345 63555 64372 131427 227840 264044 313709 336385 344858
In base 33: mu 6fb 6vb cmw mtf s3m 1lgb 1pbu 1q3m 3lml 6b78 7bfb 8o2b 9btg 9jm8

First 15 Rhonda numbers to base 34:
In base 10: 5661 14161 15620 16473 22185 37145 125579 134692 135405 138472 140369 177086 250665 255552 295614
In base 34: 4uh c8h dhe e8h j6h w4h 36lh 3ehi 3f4h 3hqo 3jeh 4h6e 6csh 6h28 7hoi

First 15 Rhonda numbers to base 35:
In base 10: 8232 9476 9633 18634 30954 41905 52215 52440 56889 61992 62146 66339 98260 102180 103305
In base 35: 6p7 7pq 7u8 f7e p9e y7a 17lu 17sa 1bfe 1fl7 1fpl 1j5e 2a7f 2def 2ebk

First 15 Rhonda numbers to base 36:
In base 10: 1000 4800 5670 8190 10998 12412 13300 15750 16821 23016 51612 52734 67744 70929 75030
In base 36: rs 3pc 4di 6bi 8hi 9ks a9g c5i cz9 hrc 13to 14ou 1g9s 1iq9 1lw6

Wren[edit]

Library: Wren-math
Library: Wren-fmt
import "./math" for Math, Int, Nums
import "./fmt" for Fmt, Conv

for (b in 2..36) {
    if (Int.isPrime(b)) continue
    var count = 0
    var rhonda = []
    var n = 1
    while (count < 15) {
        var digits = Int.digits(n, b)
        if (!digits.contains(0)) {
            if (b != 10 || (digits.contains(5) && digits.any { |d| d % 2 == 0 })) {
                var calc1 = Nums.prod(digits)
                var calc2 = b * Nums.sum(Int.primeFactors(n))
                if (calc1 == calc2) {
                    rhonda.add(n)
                    count = count + 1
                }
            }
        }
        n = n + 1
    }
    if (rhonda.count > 0) {
        System.print("\nFirst 15 Rhonda numbers in base %(b):")
        var rhonda2 = rhonda.map { |r| r.toString }.toList
        var rhonda3 = rhonda.map { |r| Conv.Itoa(r, b) }.toList
        var maxLen2 = Nums.max(rhonda2.map { |r| r.count })
        var maxLen3 = Nums.max(rhonda3.map { |r| r.count })
        var maxLen  = Math.max(maxLen2, maxLen3) + 1
        Fmt.print("In base 10:  $*s", maxLen, rhonda2)
        Fmt.print("In base $-2d:  $*s", b, maxLen, rhonda3)
    }
}
Output:
First 15 Rhonda numbers in base 4:
In base 10:        10206       11935       12150       16031       45030       94185      113022      114415      191149      244713      259753      374782      392121      503773      649902
In base 4 :      2133132     2322133     2331312     3322133    22333212   112333221   123211332   123323233   232222231   323233221   333122221  1123133332  1133232321  1322333131  2132222232

First 15 Rhonda numbers in base 6:
In base 10:       855     1029     3813     5577     7040     7304    15104    19136    35350    36992    41031    42009    60368    65536    67821
In base 6 :      3543     4433    25353    41453    52332    53452   153532   224332   431354   443132   513543   522253  1143252  1223224  1241553

First 15 Rhonda numbers in base 8:
In base 10:     1836    6318    6622   10530   14500   14739   17655   18550   25398   25956   30562   39215   39325   50875   51429
In base 8 :     3454   14256   14736   24442   34244   34623   42367   44166   61466   62544   73542  114457  114635  143273  144345

First 15 Rhonda numbers in base 9:
In base 10:    15540   21054   25331   44360   44660   44733   47652   50560   54944   76857   77142   83334   83694   96448   97944
In base 9 :    23276   31783   37665   66758   67232   67323   72326   76317   83328  126376  126733  136273  136723  156264  158316

First 15 Rhonda numbers in base 10:
In base 10:    1568   2835   4752   5265   5439   5664   5824   5832   8526  12985  15625  15698  19435  25284  25662
In base 10:    1568   2835   4752   5265   5439   5664   5824   5832   8526  12985  15625  15698  19435  25284  25662

First 15 Rhonda numbers in base 12:
In base 10:     560    800   3993   4425   4602   4888   7315   8296   9315  11849  12028  13034  14828  15052  16264
In base 12:     3A8    568   2389   2689   27B6   29B4   4297   4974   5483   6A35   6B64   7662   86B8   8864   94B4

First 15 Rhonda numbers in base 14:
In base 10:    11475   18655   20565   29631   31725   45387   58404   58667   59950   63945   67525   68904   91245   99603  125543
In base 14:     4279    6B27    76CD    AB27    B7C1   1277D   173DA   17547   17BC2   19437   1A873   1B17A   25377   28427   33A75

First 15 Rhonda numbers in base 15:
In base 10:    2392   2472  11468  15873  17424  18126  19152  20079  24388  30758  31150  33004  33550  37925  39483
In base 15:     A97    AEC   35E8   4A83   5269   5586   5A1C   5E39   735D   91A8   936A   9BA4   9E1A   B385   BA73

First 15 Rhonda numbers in base 16:
In base 10:    1000   1134   6776  15912  19624  20043  20355  23946  26296  29070  31906  32292  34236  34521  36465
In base 16:     3E8    46E   1A78   3E28   4CA8   4E4B   4F83   5D8A   66B8   718E   7CA2   7E24   85BC   86D9   8E71

First 15 Rhonda numbers in base 18:
In base 10:    1470   3000   8918  17025  19402  20650  21120  22156  26522  36549  38354  43281  46035  48768  54229
In base 18:     49C    94C   1998   2G9F   35FG   39D4   3B36   3E6G   49F8   64E9   6A6E   77A9   7G19   8696   956D

First 15 Rhonda numbers in base 20:
In base 10:    1815  11050  15295  21165  22165  30702  34510  34645  42292  44165  52059  53416  65945  78430  80712
In base 20:     4AF   17CA   1I4F   2CI5   2F85   3GF2   465A   46C5   55EC   5A85   6A2J   6DAG   84H5   9G1A   A1FC

First 15 Rhonda numbers in base 21:
In base 10:    1632   5390   8512  12992  15678  25038  29412  34017  39552  48895  49147  61376  85078  89590  91798
In base 21:     3EF    C4E    J67   189E   1EBC   2EG6   33EC   3E2I   45E9   55I7   5697   6D3E   93J7   9E34   9J37

First 15 Rhonda numbers in base 22:
In base 10:     2695    4128    7865   28800   31710   37030   71875   74306  117760  117895  121626  126002  131427  175065  192753
In base 22:      5CB     8BE     G5B    2FB2    2LB8    3AB4    6GB1    6LBC    B16G    B1CJ    B96A    BI78    C7BL    G9FB    I25B

First 15 Rhonda numbers in base 24:
In base 10:    2080   2709   3976   5628   5656   7144   8296   9030  10094  17612  20559  24616  26224  29106  31458
In base 24:     3EG    4GL    6LG    9IC    9JG    C9G    E9G    FG6    HCE   16DK   1BGF   1IHG   1LCG   22CI   26EI

First 15 Rhonda numbers in base 25:
In base 10:     6764    9633   13260   22022   53382   57640   66015   69006   97014  140130  142880  144235  159724  162565  165504
In base 25:      AKE     FA8     L5A    1A5M    3AA7    3H5F    45FF    4AA6    655E    8O55    93F5    95JA    A5DO    AA2F    AEK4

First 15 Rhonda numbers in base 26:
In base 10:     7788    9322    9374   11160   22165   27885   34905   44785   47385   49257   62517   72709   74217  108745  132302
In base 26:      BDE     DKE     DME     GD6    16KD    1F6D    1PGD    2E6D    2I2D    2KMD    3ECD    43ED    45KD    64MD    7DIE

First 15 Rhonda numbers in base 27:
In base 10:    4797  11844  12078  13200  14841  17750  24320  26883  27477  46455  52750  58581  61009  61446  61500
In base 27:     6FI    G6I    GF9    I2O    K9I    O9B   169K   19NI   1AII   29JF   2I9J   2Q9I   32IG   337L   339L

First 15 Rhonda numbers in base 28:
In base 10:    3094   5808   5832   7462  11160  13671  27270  28194  28638  39375  39550  49500  50862  52338  52938
In base 28:     3QE    7BC    7C8    9EE    E6G    HC7   16LQ   17QQ   18EM   1M67   1MCE   273O   28OE   2AL6   2BEI

First 15 Rhonda numbers in base 30:
In base 10:    3024   3168   5115   5346   5950   6762   7750   7956   8470   9476   9576   9849  10360  11495  13035
In base 30:     3AO    3FI    5KF    5S6    6IA    7FC    8IA    8P6    9CA    AFQ    AJ6    AS9    BFA    CN5    EEF

First 15 Rhonda numbers in base 32:
In base 10:    1944   3600  13520  15876  16732  16849  25410  25752  28951  47472  49610  50968  61596  64904  74005
In base 32:     1SO    3GG    D6G    FG4    GAS    GEH    OQ2    P4O    S8N   1EBG   1GEA   1HOO   1S4S   1VC8   288L

First 15 Rhonda numbers in base 33:
In base 10:      756    7040    7568   13826   24930   30613   59345   63555   64372  131427  227840  264044  313709  336385  344858
In base 33:       MU     6FB     6VB     CMW     MTF     S3M    1LGB    1PBU    1Q3M    3LML    6B78    7BFB    8O2B    9BTG    9JM8

First 15 Rhonda numbers in base 34:
In base 10:     5661   14161   15620   16473   22185   37145  125579  134692  135405  138472  140369  177086  250665  255552  295614
In base 34:      4UH     C8H     DHE     E8H     J6H     W4H    36LH    3EHI    3F4H    3HQO    3JEH    4H6E    6CSH    6H28    7HOI

First 15 Rhonda numbers in base 35:
In base 10:     8232    9476    9633   18634   30954   41905   52215   52440   56889   61992   62146   66339   98260  102180  103305
In base 35:      6P7     7PQ     7U8     F7E     P9E     Y7A    17LU    17SA    1BFE    1FL7    1FPL    1J5E    2A7F    2DEF    2EBK

First 15 Rhonda numbers in base 36:
In base 10:    1000   4800   5670   8190  10998  12412  13300  15750  16821  23016  51612  52734  67744  70929  75030
In base 36:      RS    3PC    4DI    6BI    8HI    9KS    A9G    C5I    CZ9    HRC   13TO   14OU   1G9S   1IQ9   1LW6