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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

Wolfram Mathworld - Rhonda numbers Numbers Aplenty - Rhonda numbers OEIS:A100968 - Integers n that are Rhonda numbers to base 4 OEIS:A100969 - Integers n that are Rhonda numbers to base 6 OEIS:A100970 - Integers n that are Rhonda numbers to base 8 OEIS:A100973 - Integers n that are Rhonda numbers to base 9 OEIS:A099542 - Rhonda numbers to base 10 OEIS:A100971 - Integers n that are Rhonda numbers to base 12 OEIS:A100972 - Integers n that are Rhonda numbers to base 14 OEIS:A100974 - Integers n that are Rhonda numbers to base 15 OEIS:A100975 - Integers n that are Rhonda numbers to base 16

OEIS:A255735 - Integers n that are Rhonda numbers to base 18 OEIS:A255732 - Rhonda numbers in vigesimal number system (base 20) OEIS:A255736 - Integers that are Rhonda numbers to base 30

Raku

Find and show the first 15 so as to display the namesake Rhonda number 25662. <lang perl6>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: ', ';

}</lang>

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

Library: Wren-math

Library: Wren-fmt

<lang ecmascript>import "./math" for 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| Conv.Itoa(r, b) }.toList
       var maxLen  = Nums.max(rhonda2.map { |r| r.count }) + 2
       Fmt.print("In base 10:  $*d", maxLen, rhonda)
       Fmt.print("In base $-2d:  $*s", b, maxLen, rhonda2)
   }

}</lang>

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