Pan base non-primes

Primes are prime no matter which base they are expressed in. Some numeric strings are prime in a large number of bases. (Not the same prime, but a prime.)

Pan base non-primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
For example

The numeric string "255", while obviously not a prime in base 10, is a prime in bases:

     6   8  12  14  18  24  26  32  36  38  72  84  86  92  96 102 104 108 128 134
   138 144 158 164 188 216 224 236 242 246 252 254 264 272 294 318 332 344 348 368
   374 392 396 408 428 432 446 456 468 476 482 512 522 528 542 546 552 566 572 576
   578 594 596 602 606 614 618 626 654 702 714 722 728 756 762 774 776 788 806 816
   818 822 828 836 848 864 866 872 882 888 902 908 912 918 924 932 936 942 944 956
   966 986 998

among others.

There are numeric strings however, that are not a prime in any base. Confining ourselves to 'decimal' numeric strings; the single digit numeric primes are prime in every base where they are a valid number.


E.G.

The numeric string "2" is a prime in every base except base 2, where it is invalid.

The numeric string "3" is a prime in every base except base 2 and 3, where it is invalid.

"4" is not a prime in every base except bases 2, 3, and 4 where it is an invalid number (and hence not a prime there either.)


In general, even pan-base non-primes are much more prevalent than odd, though both are fairly common.

With the exception of "10", numeric strings that end in 0 are composite in every base where they are valid.

Numeric strings where the greatest common divisor of all of the digits is more than 1 are composite in every base.

If a "decimal" numeric string N is composite in every base up to base N, it is composite in every base.

The digit 1 is an odd-ball case as it is neither prime nor composite. It typically is not included, but due to the ambiguous wording, would not be wrong if it is.


Task
  • Find and display, here on this page, the first 40 pan-base non-prime "base 10" numeric strings.
  • Find and display, here on this page, the first 20 odd pan-base non-prime "base 10" numeric strings.
  • Find and display the count of pan-base non-prime "base 10" numeric strings up to at least the numeric string "1000".
  • What percentage of them are odd / even?


See also

ALGOL 68Edit

Translation of: Wren
BEGIN # pan-base non-primes - translated from the Wren sample               #
    PR read "primes.incl.a68" PR                  # include prime utilities #
    INT    limit = 2500;

    # iterative Greatest Common Divisor routine, returns the gcd of m and n #
    PROC gcd = ( INT m, n )INT:
         BEGIN
            INT a := ABS m, b := ABS n;
            WHILE b /= 0 DO
                INT new a = b;
                b        := a MOD b;
                a        := new a
            OD;
            a
         END # gcd # ;

    # table of digit-digit Greatest Common Divisors                        #
    [ 0 : 9, 0 : 9 ]INT dd gcd;
    FOR i FROM 0 TO 9 DO FOR j FROM 0 TO 9 DO dd gcd[ i, j ] := gcd( i, j ) OD OD;

    # returns the gcd of the digits of n                                   #
    PROC gcd digits = ( INT n )INT:
         BEGIN
             STRING s  = whole( n, 0 );
             INT    g := 0;
             FOR c FROM LWB s TO UPB s DO
                 g := dd gcd[ g, ABS s[ c ] - ABS "0" ]
             OD;
             g
         END # gcd digits # ;

    # returns the number represented by s in base b                        #
    #         note s will only contain the digits 0 .. 9                   #
    PROC str to dec = ( STRING s, INT base )LONG INT:
         BEGIN
            LONG INT res := 0;
            FOR c pos FROM LWB s TO UPB s DO
                res *:= base +:= ( ABS s[ c pos ] - ABS "0" )
            OD;
            res
         END # str to dec # ;

    [ 1 : limit ]INT pbnp;
    INT pbnp count := 0;

    FOR n FROM 3 TO limit DO
       IF n MOD 10 = 0 AND n > 10 THEN
           pbnp[ pbnp count +:= 1 ] := n
       ELIF n > 9 AND gcd digits( n ) > 1 THEN
           pbnp[ pbnp count +:= 1 ] := n
       ELSE
           BOOL   comp := TRUE;
           STRING s     = whole( n, 0 );
           FOR base FROM 2 TO n WHILE comp := NOT is probably prime( str to dec( s, base ) ) DO SKIP OD;
           IF comp THEN pbnp[ pbnp count +:= 1 ] := n FI
       FI
    OD;
 
    print( ( "First 50 pan-base composites:", newline ) );
    FOR i TO IF pbnp count < 50 THEN pbnp count ELSE 50 FI DO
        print( ( " ", whole( pbnp[ i ], -3 ) ) );
        IF i MOD 10 = 0 THEN print( ( newline ) ) FI
    OD;
    print( ( newline, "First 20 odd pan-base composites:", newline ) );
    INT odd count := 0;
    FOR i TO pbnp count DO
        INT n = pbnp[ i ];
        IF ODD n THEN
            odd count +:= 1;
            IF odd count <= 20 THEN
                print( ( " ", whole( n, -3 ) ) );
                IF odd count MOD 10 = 0 THEN print( ( newline ) ) FI
            FI
        FI
    OD;

    print( ( newline
           , "Count of pan-base composites up to and including "
           , whole( limit, 0 )
           , ": "
           , whole( pbnp count, 0 )
           )
         );
    print( ( newline
           , "Number odd  = "
           , whole( odd count, 0 )
           , " or ", fixed( 100 * ( odd count / pbnp count ), -9, 6 )
           , "%"
           )
         );
    print( ( newline
           , "Number even = "
           , whole( pbnp count - odd count, 0 )
           , " or ", fixed( 100 * ( ( pbnp count - odd count ) / pbnp count ), -9, 6 )
           , "%"
           )
         )
END
Output:
First 50 pan-base composites:
   4   6   8   9  20  22  24  26  28  30
  33  36  39  40  42  44  46  48  50  55
  60  62  63  64  66  68  69  70  77  80
  82  84  86  88  90  93  96  99 100 110
 112 114 116 118 120 121 130 132 134 136

First 20 odd pan-base composites:
   9  33  39  55  63  69  77  93  99 121
 143 165 169 187 231 253 273 275 297 299

Count of pan-base composites up to and including 2500: 953
Number odd  = 161 or 16.894019%
Number even = 792 or 83.105981%

CEdit

#include <stdbool.h>
#include <stdint.h>
#include <stdio.h>

bool is_prime(uint64_t n) {
    if (n < 2)
        return false;
    if (n % 2 == 0)
        return n == 2;
    if (n % 3 == 0)
        return n == 3;
    if (n % 5 == 0)
        return n == 5;
    static const uint64_t wheel[] = {4, 2, 4, 2, 4, 6, 2, 6};
    for (uint64_t p = 7;;) {
        for (int i = 0; i < 8; ++i) {
            if (p * p > n)
                return true;
            if (n % p == 0)
                return false;
            p += wheel[i];
        }
    }
}

// Compute the digits of n in base 10, least significant digit first.
int digits(uint64_t n, uint8_t* d, int size) {
    int count = 0;
    for (; n > 0 && size > 0; n /= 10, --size, ++count)
        *d++ = n % 10;
    return count;
}

// Convert digits in the given base to a number (least significant digit first).
uint64_t from_digits(uint8_t* a, int count, uint64_t base) {
    uint64_t n = 0;
    while (count-- > 0)
        n = n * base + a[count];
    return n;
}

#define MAX_DIGITS 20

bool is_pan_base_non_prime(uint64_t n) {
    if (n < 10)
        return !is_prime(n);
    if (n > 10 && n % 10 == 0)
        return true;
    uint8_t d[MAX_DIGITS];
    int count = digits(n, d, MAX_DIGITS);
    uint8_t max_digit = 0;
    for (int i = 0; i < count; ++i) {
        if (max_digit < d[i])
            max_digit = d[i];
    }
    for (uint64_t base = max_digit + 1; base <= n; ++base) {
        if (is_prime(from_digits(d, count, base)))
            return false;
    }
    return true;
}

int main() {
    printf("First 50 prime pan-base composites:\n");
    int count = 0;
    for (uint64_t n = 2; count < 50; ++n) {
        if (is_pan_base_non_prime(n)) {
            ++count;
            printf("%3llu%c", n, count % 10 == 0 ? '\n' : ' ');
        }
    }

    printf("\nFirst 20 odd prime pan-base composites:\n");
    count = 0;
    for (uint64_t n = 3; count < 20; n += 2) {
        if (is_pan_base_non_prime(n)) {
            ++count;
            printf("%3llu%c", n, count % 10 == 0 ? '\n' : ' ');
        }
    }

    const uint64_t limit = 10000;
    int odd = 0;
    count = 0;
    for (uint64_t n = 2; n <= limit; ++n) {
        if (is_pan_base_non_prime(n)) {
            ++count;
            if (n % 2 == 1)
                ++odd;
        }
    }

    printf("\nCount of pan-base composites up to and including %llu: %d\n",
           limit, count);
    double percent = 100.0 * odd / count;
    printf("Percent odd  up to and including %llu: %f\n", limit, percent);
    printf("Percent even up to and including %llu: %f\n", limit,
           100.0 - percent);

    return 0;
}
Output:
First 50 prime pan-base composites:
  4   6   8   9  20  22  24  26  28  30
 33  36  39  40  42  44  46  48  50  55
 60  62  63  64  66  68  69  70  77  80
 82  84  86  88  90  93  96  99 100 110
112 114 116 118 120 121 130 132 134 136

First 20 odd prime pan-base composites:
  9  33  39  55  63  69  77  93  99 121
143 165 169 187 231 253 273 275 297 299

Count of pan-base composites up to and including 10000: 3849
Percent odd  up to and including 10000: 18.030657
Percent even up to and including 10000: 81.969343

C++Edit

#include <algorithm>
#include <cassert>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <numeric>
#include <vector>

bool is_prime(uint64_t n) {
    if (n < 2)
        return false;
    if (n % 2 == 0)
        return n == 2;
    if (n % 3 == 0)
        return n == 3;
    if (n % 5 == 0)
        return n == 5;
    static constexpr uint64_t wheel[] = {4, 2, 4, 2, 4, 6, 2, 6};
    for (uint64_t p = 7;;) {
        for (uint64_t w : wheel) {
            if (p * p > n)
                return true;
            if (n % p == 0)
                return false;
            p += w;
        }
    }
}

std::vector<uint64_t> digits(uint64_t n) {
    std::vector<uint64_t> d;
    for (uint64_t m = n; m > 0; m /= 10)
        d.push_back(m % 10);
    reverse(d.begin(), d.end());
    return d;
}

template <typename iterator> auto gcd(iterator begin, iterator end) {
    assert(begin != end);
    auto result = *begin++;
    for (; begin != end; ++begin)
        result = std::gcd(result, *begin);
    return result;
}

template <typename number, typename iterator>
number polyval(iterator begin, iterator end, number value) {
    number n = 0;
    for (auto i = begin; i != end; ++i)
        n = n * value + *i;
    return n;
}

bool is_pan_base_non_prime(uint64_t n) {
    if (n < 10)
        return !is_prime(n);
    if (n > 10 && n % 10 == 0)
        return true;
    auto d = digits(n);
    if (gcd(d.begin(), d.end()) > 1)
        return true;
    auto max_digit = *std::max_element(d.begin(), d.end());
    for (uint64_t base = max_digit + 1; base <= n; ++base) {
        if (is_prime(polyval(d.begin(), d.end(), base)))
            return false;
    }
    return true;
}

int main() {
    std::vector<uint64_t> pbnp;
    const uint64_t limit = 10000;
    for (uint64_t n = 2; n <= limit; ++n) {
        if (is_pan_base_non_prime(n))
            pbnp.push_back(n);
    }

    std::cout << "First 50 pan-base composites:\n";
    for (size_t i = 0; i < 50; ++i) {
        std::cout << std::setw(3) << pbnp[i]
                  << ((i + 1) % 10 == 0 ? '\n' : ' ');
    }

    std::cout << "\nFirst 20 odd pan-base composites:\n";
    for (size_t i = 0, count = 0; count < 20; ++i) {
        if (pbnp[i] % 2 == 1) {
            ++count;
            std::cout << std::setw(3) << pbnp[i]
                      << (count % 10 == 0 ? '\n' : ' ');
        }
    }

    size_t total = pbnp.size();
    size_t odd = std::count_if(pbnp.begin(), pbnp.end(),
                               [](uint64_t n) { return n % 2 == 1; });
    std::cout << "\nCount of pan-base composites up to and including " << limit
              << ": " << total << '\n';
    auto percent = (100.0 * odd) / total;
    std::cout << std::fixed;
    std::cout << "Percent odd  up to and including " << limit << ": " << percent
              << '\n';
    std::cout << "Percent even up to and including " << limit << ": "
              << 100.0 - percent << '\n';
}
Output:
First 50 pan-base composites:
  4   6   8   9  20  22  24  26  28  30
 33  36  39  40  42  44  46  48  50  55
 60  62  63  64  66  68  69  70  77  80
 82  84  86  88  90  93  96  99 100 110
112 114 116 118 120 121 130 132 134 136

First 20 odd pan-base composites:
  9  33  39  55  63  69  77  93  99 121
143 165 169 187 231 253 273 275 297 299

Count of pan-base composites up to and including 10000: 3849
Percent odd  up to and including 10000: 18.030657
Percent even up to and including 10000: 81.969343

JEdit

Implementation:
pbnp=: {{ if. 10 > y do. -.1 p: y return. end.
  digits=. 10 #.inv y
  */0=1 p: ((>./digits)+i.y) #."0 1 digits
}}"0
Task examples:
   40{.1+I.pbnp 1+i.1e3 NB. first 40 pan based non primes
1 4 6 8 9 20 22 24 26 28 30 33 36 39 40 42 44 46 48 50 55 60 62 63 64 66 68 69 70 77 80 82 84 86 88 90 93 96 99 100
   20{.(#~ 2&|)1+I.pbnp 1+i.1e3 NB. first 20 odd pan based non primes
1 9 33 39 55 63 69 77 93 99 121 143 165 169 187 231 253 273 275 297
   #1+I.pbnp 1+i.1e3  NB. number of pan based non primes up to 1000
378
   #(#~ 2&|)1+I.pbnp 1+i.1e3 NB. number of odd pan based non primes up to 1000
64
   100*(+/%#)2|1+I.pbnp 1+i.1e3 NB. percent odd pan based non primes up to 1000
16.9761

JavaEdit

public class PanBaseNonPrimes {
    public static void main(String[] args) {
        System.out.printf("First 50 prime pan-base composites:\n");
        int count = 0;
        for (long n = 2; count < 50; ++n) {
            if (isPanBaseNonPrime(n)) {
                ++count;
                System.out.printf("%3d%c", n, count % 10 == 0 ? '\n' : ' ');
            }
        }
    
        System.out.printf("\nFirst 20 odd prime pan-base composites:\n");
        count = 0;
        for (long n = 3; count < 20; n += 2) {
            if (isPanBaseNonPrime(n)) {
                ++count;
                System.out.printf("%3d%c", n, count % 10 == 0 ? '\n' : ' ');
            }
        }
    
        final long limit = 10000;
        count = 0;
        int odd = 0;
        for (long n = 2; n <= limit; ++n) {
            if (isPanBaseNonPrime(n)) {
                ++count;
                if (n % 2 == 1)
                    ++odd;
            }
        }
    
        System.out.printf("\nCount of pan-base composites up to and including %d: %d\n",
                            limit, count);
        double percent = 100.0 * odd / count;
        System.out.printf("Percent odd  up to and including %d: %f\n",
                            limit, percent);
        System.out.printf("Percent even up to and including %d: %f\n",
                            limit, 100.0 - percent);
    }

    private static boolean isPanBaseNonPrime(long n) {
        if (n < 10)
            return !isPrime(n);
        if (n > 10 && n % 10 == 0)
            return true;
        byte[] d = new byte[20];
        int count = digits(n, d);
        byte max_digit = 0;
        for (int i = 0; i < count; ++i) {
            if (max_digit < d[i])
                max_digit = d[i];
        }
        for (long base = max_digit + 1; base <= n; ++base) {
            if (isPrime(fromDigits(d, count, base)))
                return false;
        }
        return true;
    }

    private static final long[] WHEEL = {4, 2, 4, 2, 4, 6, 2, 6};

    private static boolean isPrime(long n) {
        if (n < 2)
            return false;
        if (n % 2 == 0)
            return n == 2;
        if (n % 3 == 0)
            return n == 3;
        if (n % 5 == 0)
            return n == 5;
        for (long p = 7;;) {
            for (int i = 0; i < 8; ++i) {
                if (p * p > n)
                    return true;
                if (n % p == 0)
                    return false;
                p += WHEEL[i];
            }
        }
    }

    // Compute the digits of n in base 10, least significant digit first.
    private static int digits(long n, byte[] d) {
        int count = 0;
        for (; n > 0 && count < d.length; n /= 10, ++count)
            d[count] = (byte)(n % 10);
        return count;
    }

    // Convert digits in the given base to a number (least significant digit first).
    private static long fromDigits(byte[] a, int count, long base) {
        long n = 0;
        while (count-- > 0)
            n = n * base + a[count];
        return n;
    }
}
Output:
First 50 prime pan-base composites:
  4   6   8   9  20  22  24  26  28  30
 33  36  39  40  42  44  46  48  50  55
 60  62  63  64  66  68  69  70  77  80
 82  84  86  88  90  93  96  99 100 110
112 114 116 118 120 121 130 132 134 136

First 20 odd prime pan-base composites:
  9  33  39  55  63  69  77  93  99 121
143 165 169 187 231 253 273 275 297 299

Count of pan-base composites up to and including 10000: 3849
Percent odd  up to and including 10000: 18.030657
Percent even up to and including 10000: 81.969343

JuliaEdit

using Primes

ispanbasecomposite(n) = (d = digits(n); all(b -> !isprime(evalpoly(b, d)), maximum(d)+1:max(10, n)))

panbase2500 = filter(ispanbasecomposite, 2:2500)
oddpanbase2500 = filter(isodd, panbase2500)
ratio = length(oddpanbase2500) // length(panbase2500)

println("First 50 pan base non-primes:")
foreach(p -> print(lpad(p[2], 4), p[1] % 10 == 0 ? "\n" : ""), pairs(panbase2500[1:50]))

println("\nFirst 20 odd pan base non-primes:")
foreach(p -> print(lpad(p[2], 4), p[1] % 10 == 0 ? "\n" : ""), pairs(oddpanbase2500[1:20]))

println("\nCount of pan-base composites up to and including 2500: ", length(panbase2500))

println("Odd up to and including 2500: ", ratio, ", or ", Float16(ratio * 100), "%.")
println("Even up to and including 2500: ", 1 - ratio, ", or ", Float16((1.0 - ratio) * 100), "%.")
Output:
First 50 pan base non-primes:
   4   6   8   9  20  22  24  26  28  30
  33  36  39  40  42  44  46  48  50  55
  60  62  63  64  66  68  69  70  77  80
  82  84  86  88  90  93  96  99 100 110
 112 114 116 118 120 121 130 132 134 136

First 20 odd pan base non-primes:       
   9  33  39  55  63  69  77  93  99 121
 143 165 169 187 231 253 273 275 297 299

Count of pan-base composites up to and including 2500: 953
Odd up to and including 2500: 161//953, or 16.89%.
Even up to and including 2500: 792//953, or 83.1%.

PascalEdit

Free PascalEdit

program PanBaseNonPrime;
  // Check Pan-Base Non-Prime
  {$IFDEF FPC}{$MODE DELPHI}{$OPTIMIZATION ON,ALL}{$ENDIF}
  {$IFDEF WINDOWS}{$APPTYPE CONSOLE}{$ENDIF}
// MAXLIMIT beyond 10000 gets really slow 5 digits, depends on isPrime
//10004 checked til base 10003 -> 10003⁴+3 = 1.0012e16, takes >1 s longer
//real   0m1,307s
// 9999 checked til base  9998 -> 8,99555E12 much smaller
//real   0m0,260s
type
  tDgts = 0..31;// Int32 is faster than 0..9 -> word
  tUsedDgts = set of tDgts;
  tDecDigits = packed record
                 decdgts   :array[0..20] of byte;
                 decmaxIdx :byte;
                 decmaxDgt :byte;
                 decUsedDgt :tUsedDgts;
               end;
const
  MAXLIMIT = 2500;
  WithGCDNotOne : array[0..24] of tUsedDgts =
      //all the same digits
     ([0],[2],[3],[4],[5],[6],[7],[8],[9],
      //all even
      [2,4],[2,6],[2,8],
      [2,4,6],[2,4,8],[2,6,8],
      [2,4,6,8],
      [4,6],[4,8],
      [4,6,8],[2,4,6,8],
      [6,8],
      //all divible 3
      [3,6],[3,9],
      [3,6,9],
      [6,9]);
var
  gblCnt,
  gblOddCnt :NativeINt;

procedure OutDecDigits(var Dgts:tDecDigits);
var
  idx : nativeInt;
begin
  with Dgts do
  begin
    idx := decMaxIDx;
    repeat
      dec(idx);
      write(decdgts[idx]);
    until idx <= 0;
    write(decmaxdgt:3);
    writeln;
  end;
end;

procedure CountOne(n:NativeInt);inline;
Begin
  inc(gblCnt);
  If odd(n) then
    inc(gblOddCnt);
end;

procedure OutCountOne(n:NativeInt);
begin
  CountOne(n);
  write(n:5);
  if gblCnt mod 10 = 0 then
    writeln;
end;

function CheckGCD(var Dgts:tDecDigits):boolean;
var
  idx: NativeInt;
  UsedDgts:tUsedDgts;
begin
  UsedDgts := Dgts.decUsedDgt;
  For idx := Low(WithGCDNotOne) to High(WithGCDNotOne) do
    if UsedDgts = WithGCDNotOne[idx] then
      Exit(true);
  Exit(false);
end;

procedure ConvToDecDgt(n : NativeUint;out Dgts:tDecDigits);//inline;
var
  dgt,maxdgt,idx,q :NativeInt;
  UsedDgts : tUsedDgts;
begin
  UsedDgts := [];
  maxdgt := 0;
  idx := 0;
  repeat
    q := n div 10;
    dgt := n-q*10;
    Dgts.decdgts[idx]:= dgt;
    include(UsedDgts,dgt);
    IF maxdgt<dgt then
      maxdgt := dgt;
    inc(idx);
    n := q;
  until n = 0;

  with Dgts do
  Begin
    decMaxIDx := idx;
    decMaxdgt := maxDgt;
    decUsedDgt := UsedDgts;
  end;
end;

function ConvDgtToBase(var Dgts:tDecDigits;base:NativeInt):NativeUInt;
var
  idx :NativeInt;
begin
  result := 0;
  if base<= Dgts.decMaxdgt then
    EXIT;

  with Dgts do
  Begin
    idx := decMaxIDx;
    repeat
      dec(idx);
      result := result*base+decdgts[idx];
    until idx <= 0;
  end;
end;

function isPrime(n: NativeInt):boolean;
//simple trial division
var
  j : nativeInt;
begin
  if n in [2,3,5,7,11,13,17,19,23,29,31] then
    EXIT(true);
  if n<32 then
    EXIT(false);
  if not(odd(n)) then
    EXIT(false);
  if n mod 3 = 0 then
    EXIT(false);
  if n mod 5 = 0 then
    EXIT(false);
  j := 7;
  while j*j<=n do
  begin
    if n mod j = 0 then
      EXIT(false);
    inc(j,4);
    if n mod j = 0 then
      EXIT(false);
    inc(j,2);
  end;
  EXIT(true);
end;

function CheckPanBaseNonPrime(n: NativeUint):boolean;
var
  myDecDgts:tDecDigits;
  b,num : NativeInt;
Begin
  result := true;
  ConvToDecDgt(n,myDecDgts);
  if (n>10) then
  Begin
    if (myDecDgts.decdgts[0] = 0) then
      Exit;
    if CheckGCD(myDecDgts) then
      Exit;
  end;

  b := myDecDgts.decmaxdgt+1;
  if b >= n then
  Begin
    if isPrime(n) then
      Exit(false);
  end
  else
  begin
    while b < n do
    begin
      num := ConvDgtToBase(myDecDgts,b);
      if isPrime(num) then
        EXIT(false);
      inc(b);
    end;
  end;
end;
var
  i : NativeInt;

BEGIN
  writeln('First 50 pan-base non-prime numbers ');
  gblCnt := 0;
  gblOddCnt := 0;
  For i := 3 to MAXLIMIT do
  Begin
    if CheckPanBaseNonPrime(i) then
      OutCountOne(i);
    if gblCnt = 50 then
      break;
  end;
  writeln;

  writeln('First 20 pan-base non-prime odd numbers ');
  gblCnt := 0;
  gblOddCnt := 0;
  For i := 3 to MAXLIMIT do
  Begin
    if ODD(i) then
    Begin
      if CheckPanBaseNonPrime(i) then
         OutCountOne(i);
      if gblOddCnt = 20 then
        break;
    end;
  end;
  writeln;

  gblCnt := 0;
  gblOddCnt := 0;
  For i := 3 to MAXLIMIT do
    if CheckPanBaseNonPrime(i) then
      CountOne(i);
  writeln('Count of pan-base composites up to and including ',MAXLIMIT,' : ',gblCnt);
  writeln('odd  up to and including ',MAXLIMIT,' = ',gblOddCnt:4,' equals ',gblOddCnt/gblCnt*100:10:6,'%');
  writeln('even up to and including ',MAXLIMIT,' = ',gblCnt-gblOddCnt:4,' equals ',(gblCnt-gblOddCnt)/gblCnt*100:10:6,'%');

END.
Output:
First 50 pan-base non-prime numbers
    4    6    8    9   20   22   24   26   28   30
   33   36   39   40   42   44   46   48   50   55
   60   62   63   64   66   68   69   70   77   80
   82   84   86   88   90   93   96   99  100  110
  112  114  116  118  120  121  130  132  134  136

First 20 pan-base non-prime odd numbers
    9   33   39   55   63   69   77   93   99  121
  143  165  169  187  231  253  273  275  297  299

Count of pan-base composites up to and including 2500 : 953
odd  up to and including 2500 =  161 equals  16.894019%
even up to and including 2500 =  792 equals  83.105981%

PerlEdit

Library: ntheory
use v5.36;
use ntheory <fromdigits is_prime>;
use List::AllUtils <max firstidx>;

sub table ($c, @V) { my $t = $c * (my $w = 2 + length max @V); ( sprintf( ('%'.$w.'d')x@V, @V) ) =~ s/.{1,$t}\K/\n/gr }

my $max = 2500;
my @np = <4 6 8 9>;
for my $n (11..$max) {
    push @np, $n unless max map { max(split '',$n) < $_ and is_prime fromdigits($n,$_) } 2..$n;
}

say "First 50 pan-base composites:\n"                        . table 10, @np[0..49];
say "First 20 odd pan-base composites:\n"                    . table 10, (grep { 0 !=  $_ % 2 } @np)[0..19];
say "Count of pan-base composites up to and including $max: ". (my $f = 1 + firstidx { $max <= $_ } @np);
say "Percent odd  up to and including $max: "                . sprintf '%.3f', 100 * (grep { 0 != $_ % 2 } @np[0..$f-1]) / $f;
say "Percent even up to and including $max: "                . sprintf '%.3f', 100 * (grep { 0 == $_ % 2 } @np[0..$f-1]) / $f;
Output:
First 50 pan-base composites:
    4    6    8    9   20   22   24   26   28   30
   33   36   39   40   42   44   46   48   50   55
   60   62   63   64   66   68   69   70   77   80
   82   84   86   88   90   93   96   99  100  110
  112  114  116  118  120  121  130  132  134  136

First 20 odd pan-base composites:
    9   33   39   55   63   69   77   93   99  121
  143  165  169  187  231  253  273  275  297  299

Count of pan-base composites up to and including 2500: 953
Percent odd  up to and including 2500: 16.894
Percent even up to and including 2500: 83.106

PhixEdit

Translation of: Wren
with javascript_semantics
constant lim = 2500
sequence pbnp = {}
for n=3 to lim do
    sequence digits = sq_sub(sprintf("%d",n),'0')
    if (remainder(n,10)=0 and n>10)
    or (n>9 and gcd(digits)>1) then
        pbnp &= n
    else    
        bool composite = true
        for base=2 to n do
            atom d = 0
            for c in digits do
                d = d*base + c
            end for
            if is_prime(d) then
                composite = false
                exit
            end if
        end for
        if composite then pbnp &= n end if
    end if
end for
 
sequence odds = filter(pbnp,odd)
integer tc = length(pbnp),
        oc = length(odds),
        ec = tc-oc
string f50 = join_by(pbnp[1..50],1,10," ",fmt:="%3d"),
       o20 = join_by(odds[1..20],1,10," ",fmt:="%3d")
printf(1,"First 50 pan-base composites:\n%s\n",f50)
printf(1,"First 20 odd pan-base composites:\n%s\n",o20)
printf(1,"Count of pan-base composites up to and including %d: %d\n",{lim,tc})
printf(1,"Number odd  = %3d or %9.6f%%\n", {oc,oc/tc*100})
printf(1,"Number even = %3d or %9.6f%%\n", {ec,ec/tc*100})

Output same as Wren

RakuEdit

use Base::Any;
use List::Divvy;

my @np = 4,6,8,9, |lazy (11..*).hyper.grep( -> $n { ($n.substr(*-1) eq '0') || (1 < [gcd] $n.comb».Int) || none (2..$n).map: { try "$n".&from-base($_).is-prime } } );

put "First 50 pan-base composites:\n" ~ @np[^50].batch(10)».fmt("%3s").join: "\n";
put "\nFirst 20 odd pan-base composites:\n" ~ @np.grep(* % 2)[^20].batch(10)».fmt("%3s").join: "\n";

my $threshold = 2500;
put "\nCount of pan-base composites up to and including $threshold: " ~ +@np.&upto($threshold);

put "Percent odd  up to and including $threshold: " ~ +@np.&upto($threshold).grep(* % 2) / +@np.&upto($threshold) × 100;
put "Percent even up to and including $threshold: " ~ +@np.&upto($threshold).grep(* %% 2) / +@np.&upto($threshold) × 100;
Output:
First 50 pan-base composites:
  4   6   8   9  20  22  24  26  28  30
 33  36  39  40  42  44  46  48  50  55
 60  62  63  64  66  68  69  70  77  80
 82  84  86  88  90  93  96  99 100 110
112 114 116 118 120 121 130 132 134 136

First 20 odd pan-base composites:
  9  33  39  55  63  69  77  93  99 121
143 165 169 187 231 253 273 275 297 299

Count of pan-base composites up to and including 2500: 953
Percent odd  up to and including 2500: 16.894019
Percent even up to and including 2500: 83.105981

WrenEdit

Library: Wren-math
Library: Wren-fmt
import "./math" for Int
import "./fmt" for Fmt

var strToDec = Fn.new { |s, b|
    var res =  0
    for (c in s) {
        var d = Num.fromString(c)
        res = res * b + d
    }
    return res
}

var limit = 2500
var pbnp = []
for (n in 3..limit) {
    if (n % 10 == 0 && n > 10) {
        pbnp.add(n)
    } else if (n > 9 && Int.gcd(Int.digits(n)) > 1) {
        pbnp.add(n)
    } else {
        var comp = true
        for (b in 2...n) {
            var d = strToDec.call(n.toString, b)
            if (Int.isPrime(d)) {
                comp = false
                break
            }
        }
        if (comp) pbnp.add(n)
    }
}

System.print("First 50 pan-base composites:")
Fmt.tprint("$3d", pbnp[0..49], 10)

System.print("\nFirst 20 odd pan-base composites:")
var odd = pbnp.where { |n| n % 2 == 1 }.toList
Fmt.tprint("$3d", odd[0..19], 10)

var tc
System.print("\nCount of pan-base composites up to and including %(limit): %(tc = pbnp.count)")
var c
Fmt.print("Number odd  = $3d or $9.6f\%", c = odd.count, c/tc * 100)
Fmt.print("Number even = $3d or $9.6f\%", c = tc - c, c/tc * 100)
Output:
First 50 pan-base composites:
  4   6   8   9  20  22  24  26  28  30 
 33  36  39  40  42  44  46  48  50  55 
 60  62  63  64  66  68  69  70  77  80 
 82  84  86  88  90  93  96  99 100 110 
112 114 116 118 120 121 130 132 134 136 

First 20 odd pan-base composites:
  9  33  39  55  63  69  77  93  99 121 
143 165 169 187 231 253 273 275 297 299 

Count of pan-base composites up to and including 2500: 953
Number odd  = 161 or 16.894019%
Number even = 792 or 83.105981%