Fermat pseudoprimes

A Fermat pseudoprime is a positive composite integer that passes the Fermat primality test.

Fermat's little theorem states that if p is prime and a is coprime to p, then a^p−1 − 1 is divisible by p.

For an integer a > 1, if a composite integer x divides a^x−1 − 1, then x is called a Fermat pseudoprime to base a.

Fermat pseudoprimes to base 2 are sometimes called Sarrus numbers or Poulet numbers.

Fermat pseudoprimes can be found to any positive integer base. When using a base integer a = 1, this method returns all composite numbers.

Task

For base integers a of 1 through 20:

Find the count of pseudoprimes up to and including 12,000.
Show the first 20 pseudoprimes.

Stretch

Extend the count threshold out to 25,000, 50,000 or higher.

See also

Wikipedia: Fermat pseudoprime OEIS:A002808 - Composite numbers OEIS:A001567 - Fermat pseudoprimes to base 2 OEIS:A005935 - Fermat pseudoprimes to base 3 OEIS:A020136 - Fermat pseudoprimes to base 4 OEIS:A005936 - Fermat pseudoprimes to base 5 OEIS:A005937 - Fermat pseudoprimes to base 6 OEIS:A005938 - Fermat pseudoprimes to base 7 OEIS:A020137 - Fermat pseudoprimes to base 8 OEIS:A020138 - Fermat pseudoprimes to base 9 OEIS:A005939 - Fermat pseudoprimes to base 10 OEIS:A020139 - Fermat pseudoprimes to base 11 OEIS:A020140 - Fermat pseudoprimes to base 12 OEIS:A020141 - Fermat pseudoprimes to base 13 OEIS:A020142 - Fermat pseudoprimes to base 14 OEIS:A020143 - Fermat pseudoprimes to base 15 OEIS:A020144 - Fermat pseudoprimes to base 16 OEIS:A020145 - Fermat pseudoprimes to base 17 OEIS:A020146 - Fermat pseudoprimes to base 18 OEIS:A020147 - Fermat pseudoprimes to base 19 OEIS:A020148 - Fermat pseudoprimes to base 20

Realated Task Modular exponentiation

F#

This task uses Extensible Prime Generator (F#) <lang fsharp> // Fermat pseudoprimes. Nigel Galloway: August 17th., 2022 let fp(a:int)=let a=bigint a in primesI()|>Seq.pairwise|>Seq.collect(fun(n,g)->seq{for n in n+1I..g-1I do if bigint.ModPow(a,n-1I,n)=1I then yield n}) {1..20}|>Seq.iter(fun n->printf $"Base %2d{n} - Up to 50000: %5d{fp n|>Seq.takeWhile((>=)50000I)|>Seq.length} First 2O: ("; fp n|>Seq.take 20|>Seq.iter(printf "%A "); printfn ")") </lang>

Output:

Base  1 - Up to 50000: 44866 First 2O: (4 6 8 9 10 12 14 15 16 18 20 21 22 24 25 26 27 28 30 32 )
Base  2 - Up to 50000:    55 First 2O: (341 561 645 1105 1387 1729 1905 2047 2465 2701 2821 3277 4033 4369 4371 4681 5461 6601 7957 8321 )
Base  3 - Up to 50000:    53 First 2O: (91 121 286 671 703 949 1105 1541 1729 1891 2465 2665 2701 2821 3281 3367 3751 4961 5551 6601 )
Base  4 - Up to 50000:   111 First 2O: (15 85 91 341 435 451 561 645 703 1105 1247 1271 1387 1581 1695 1729 1891 1905 2047 2071 )
Base  5 - Up to 50000:    54 First 2O: (4 124 217 561 781 1541 1729 1891 2821 4123 5461 5611 5662 5731 6601 7449 7813 8029 8911 9881 )
Base  6 - Up to 50000:    74 First 2O: (35 185 217 301 481 1105 1111 1261 1333 1729 2465 2701 2821 3421 3565 3589 3913 4123 4495 5713 )
Base  7 - Up to 50000:    49 First 2O: (6 25 325 561 703 817 1105 1825 2101 2353 2465 3277 4525 4825 6697 8321 10225 10585 10621 11041 )
Base  8 - Up to 50000:   150 First 2O: (9 21 45 63 65 105 117 133 153 231 273 341 481 511 561 585 645 651 861 949 )
Base  9 - Up to 50000:   113 First 2O: (4 8 28 52 91 121 205 286 364 511 532 616 671 697 703 946 949 1036 1105 1288 )
Base 10 - Up to 50000:    65 First 2O: (9 33 91 99 259 451 481 561 657 703 909 1233 1729 2409 2821 2981 3333 3367 4141 4187 )
Base 11 - Up to 50000:    61 First 2O: (10 15 70 133 190 259 305 481 645 703 793 1105 1330 1729 2047 2257 2465 2821 4577 4921 )
Base 12 - Up to 50000:    91 First 2O: (65 91 133 143 145 247 377 385 703 1045 1099 1105 1649 1729 1885 1891 2041 2233 2465 2701 )
Base 13 - Up to 50000:    68 First 2O: (4 6 12 21 85 105 231 244 276 357 427 561 1099 1785 1891 2465 2806 3605 5028 5149 )
Base 14 - Up to 50000:    69 First 2O: (15 39 65 195 481 561 781 793 841 985 1105 1111 1541 1891 2257 2465 2561 2665 2743 3277 )
Base 15 - Up to 50000:    42 First 2O: (14 341 742 946 1477 1541 1687 1729 1891 1921 2821 3133 3277 4187 6541 6601 7471 8701 8911 9073 )
Base 16 - Up to 50000:   145 First 2O: (15 51 85 91 255 341 435 451 561 595 645 703 1105 1247 1261 1271 1285 1387 1581 1687 )
Base 17 - Up to 50000:    63 First 2O: (4 8 9 16 45 91 145 261 781 1111 1228 1305 1729 1885 2149 2821 3991 4005 4033 4187 )
Base 18 - Up to 50000:    98 First 2O: (25 49 65 85 133 221 323 325 343 425 451 637 931 1105 1225 1369 1387 1649 1729 1921 )
Base 19 - Up to 50000:    93 First 2O: (6 9 15 18 45 49 153 169 343 561 637 889 905 906 1035 1105 1629 1661 1849 1891 )
Base 20 - Up to 50000:    66 First 2O: (21 57 133 231 399 561 671 861 889 1281 1653 1729 1891 2059 2413 2501 2761 2821 2947 3059 )
Real: 00:00:00.632

Julia

<lang ruby>using Primes

ispseudo(n, base) = !isprime(n) && BigInt(base)^(n - 1) % n == 1

for b in 1:20

   pseudos = filter(n -> ispseudo(n, b), 1:50000)
   println("Base ", lpad(b, 2), " up to 50000: ", lpad(length(pseudos), 5), "  First 20: ", pseudos[1:20])

end

</lang>

Output:

Base  1 up to 50000: 44866  First 20: [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32]
Base  2 up to 50000:    55  First 20: [341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321]      
Base  3 up to 50000:    53  First 20: [91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601]
Base  4 up to 50000:   111  First 20: [15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071]
Base  5 up to 50000:    54  First 20: [4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881]
Base  6 up to 50000:    74  First 20: [35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729, 2465, 2701, 2821, 3421, 3565, 3589, 3913, 4123, 4495, 5713]
Base  7 up to 50000:    49  First 20: [6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321, 10225, 10585, 10621, 11041]
Base  8 up to 50000:   150  First 20: [9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949]
Base  9 up to 50000:   113  First 20: [4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288]
Base 10 up to 50000:    65  First 20: [9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187]
Base 11 up to 50000:    61  First 20: [10, 15, 70, 133, 190, 259, 305, 481, 645, 703, 793, 1105, 1330, 1729, 2047, 2257, 2465, 2821, 4577, 4921]
Base 12 up to 50000:    91  First 20: [65, 91, 133, 143, 145, 247, 377, 385, 703, 1045, 1099, 1105, 1649, 1729, 1885, 1891, 2041, 2233, 2465, 2701]
Base 13 up to 50000:    68  First 20: [4, 6, 12, 21, 85, 105, 231, 244, 276, 357, 427, 561, 1099, 1785, 1891, 2465, 2806, 3605, 5028, 5149]
Base 14 up to 50000:    69  First 20: [15, 39, 65, 195, 481, 561, 781, 793, 841, 985, 1105, 1111, 1541, 1891, 2257, 2465, 2561, 2665, 2743, 3277]
Base 15 up to 50000:    42  First 20: [14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921, 2821, 3133, 3277, 4187, 6541, 6601, 7471, 8701, 8911, 9073]
Base 16 up to 50000:   145  First 20: [15, 51, 85, 91, 255, 341, 435, 451, 561, 595, 645, 703, 1105, 1247, 1261, 1271, 1285, 1387, 1581, 1687]
Base 17 up to 50000:    63  First 20: [4, 8, 9, 16, 45, 91, 145, 261, 781, 1111, 1228, 1305, 1729, 1885, 2149, 2821, 3991, 4005, 4033, 4187]
Base 18 up to 50000:    98  First 20: [25, 49, 65, 85, 133, 221, 323, 325, 343, 425, 451, 637, 931, 1105, 1225, 1369, 1387, 1649, 1729, 1921]
Base 19 up to 50000:    93  First 20: [6, 9, 15, 18, 45, 49, 153, 169, 343, 561, 637, 889, 905, 906, 1035, 1105, 1629, 1661, 1849, 1891]
Base 20 up to 50000:    66 First 20: [21, 57, 133, 231, 399, 561, 671, 861, 889, 1281, 1653, 1729, 1891, 2059, 2413, 2501, 2761, 2821, 2947, 3059]

Phix

with javascript_semantics
include mpfr.e 

function fermat_pseudoprime(integer x, base)
    if is_prime(x) then return false end if
    mpz z = mpz_init(x),
        a = mpz_init(base)
    mpz_powm_ui(z, a, x-1, z)
    return mpz_cmp_si(z,1) == 0
end function
 
sequence limits = {12000, 25000, 50000, 100000}
for base=1 to 20 do
    integer count = 0, nlx = 1, nl = limits[1]
    sequence first20 = {}, counts = repeat(0,length(limits))
    for x=2 to limits[$] do
        if fermat_pseudoprime(x, base) then
            if count<20 then first20 &= x end if
            count += 1
        end if
        if x=nl then
            counts[nlx] = count
            if nlx<length(limits) then
                nlx += 1
                nl = limits[nlx]
            end if
        end if
    end for
    string s = join(shorten(first20,"",3,"%d"),","),
           c = join(columnize({limits,counts}),", ",fmt:="%d:%5d")
    printf(1,"Base %2d, counts to %s, first 20: %s\n", {base, c, s})
end for

Output:

Base  1, counts to 12000:10561, 25000:22237, 50000:44866, 100000:90407, first 20: 4,6,8,...,28,30,32
Base  2, counts to 12000:   25, 25000:   38, 50000:   55, 100000:   78, first 20: 341,561,645,...,6601,7957,8321
Base  3, counts to 12000:   25, 25000:   38, 50000:   53, 100000:   78, first 20: 91,121,286,...,4961,5551,6601
Base  4, counts to 12000:   50, 25000:   75, 50000:  111, 100000:  153, first 20: 15,85,91,...,1905,2047,2071
Base  5, counts to 12000:   22, 25000:   35, 50000:   54, 100000:   73, first 20: 4,124,217,...,8029,8911,9881
Base  6, counts to 12000:   31, 25000:   46, 50000:   74, 100000:  104, first 20: 35,185,217,...,4123,4495,5713
Base  7, counts to 12000:   21, 25000:   32, 50000:   49, 100000:   73, first 20: 6,25,325,...,10585,10621,11041
Base  8, counts to 12000:   76, 25000:  110, 50000:  150, 100000:  218, first 20: 9,21,45,...,651,861,949
Base  9, counts to 12000:   55, 25000:   83, 50000:  113, 100000:  164, first 20: 4,8,28,...,1036,1105,1288
Base 10, counts to 12000:   35, 25000:   53, 50000:   65, 100000:   90, first 20: 9,33,91,...,3367,4141,4187
Base 11, counts to 12000:   30, 25000:   41, 50000:   61, 100000:   89, first 20: 10,15,70,...,2821,4577,4921
Base 12, counts to 12000:   35, 25000:   60, 50000:   91, 100000:  127, first 20: 65,91,133,...,2233,2465,2701
Base 13, counts to 12000:   31, 25000:   51, 50000:   68, 100000:   91, first 20: 4,6,12,...,3605,5028,5149
Base 14, counts to 12000:   33, 25000:   51, 50000:   69, 100000:   96, first 20: 15,39,65,...,2665,2743,3277
Base 15, counts to 12000:   22, 25000:   31, 50000:   42, 100000:   70, first 20: 14,341,742,...,8701,8911,9073
Base 16, counts to 12000:   69, 25000:  102, 50000:  145, 100000:  200, first 20: 15,51,85,...,1387,1581,1687
Base 17, counts to 12000:   31, 25000:   44, 50000:   63, 100000:   83, first 20: 4,8,9,...,4005,4033,4187
Base 18, counts to 12000:   46, 25000:   69, 50000:   98, 100000:  134, first 20: 25,49,65,...,1649,1729,1921
Base 19, counts to 12000:   48, 25000:   70, 50000:   93, 100000:  121, first 20: 6,9,15,...,1661,1849,1891
Base 20, counts to 12000:   35, 25000:   49, 50000:   66, 100000:  101, first 20: 21,57,133,...,2821,2947,3059

Raku

<lang perl6>use List::Divvy; for 1..20 -> $base {

   my @pseudo = lazy (2..*).hyper.grep: { !.is-prime && (1 == expmod $base, $_ - 1, $_) }
   my $threshold = 100_000;
   say $base.fmt("Base %2d - Up to $threshold: ") ~ (+@pseudo.&upto: $threshold).fmt('%5d')
       ~ "  First 20: " ~ @pseudo[^20].gist

}</lang>

Base  1 - Up to 100000: 90407  First 20: (4 6 8 9 10 12 14 15 16 18 20 21 22 24 25 26 27 28 30 32)
Base  2 - Up to 100000:    78  First 20: (341 561 645 1105 1387 1729 1905 2047 2465 2701 2821 3277 4033 4369 4371 4681 5461 6601 7957 8321)
Base  3 - Up to 100000:    78  First 20: (91 121 286 671 703 949 1105 1541 1729 1891 2465 2665 2701 2821 3281 3367 3751 4961 5551 6601)
Base  4 - Up to 100000:   153  First 20: (15 85 91 341 435 451 561 645 703 1105 1247 1271 1387 1581 1695 1729 1891 1905 2047 2071)
Base  5 - Up to 100000:    73  First 20: (4 124 217 561 781 1541 1729 1891 2821 4123 5461 5611 5662 5731 6601 7449 7813 8029 8911 9881)
Base  6 - Up to 100000:   104  First 20: (35 185 217 301 481 1105 1111 1261 1333 1729 2465 2701 2821 3421 3565 3589 3913 4123 4495 5713)
Base  7 - Up to 100000:    73  First 20: (6 25 325 561 703 817 1105 1825 2101 2353 2465 3277 4525 4825 6697 8321 10225 10585 10621 11041)
Base  8 - Up to 100000:   218  First 20: (9 21 45 63 65 105 117 133 153 231 273 341 481 511 561 585 645 651 861 949)
Base  9 - Up to 100000:   164  First 20: (4 8 28 52 91 121 205 286 364 511 532 616 671 697 703 946 949 1036 1105 1288)
Base 10 - Up to 100000:    90  First 20: (9 33 91 99 259 451 481 561 657 703 909 1233 1729 2409 2821 2981 3333 3367 4141 4187)
Base 11 - Up to 100000:    89  First 20: (10 15 70 133 190 259 305 481 645 703 793 1105 1330 1729 2047 2257 2465 2821 4577 4921)
Base 12 - Up to 100000:   127  First 20: (65 91 133 143 145 247 377 385 703 1045 1099 1105 1649 1729 1885 1891 2041 2233 2465 2701)
Base 13 - Up to 100000:    91  First 20: (4 6 12 21 85 105 231 244 276 357 427 561 1099 1785 1891 2465 2806 3605 5028 5149)
Base 14 - Up to 100000:    96  First 20: (15 39 65 195 481 561 781 793 841 985 1105 1111 1541 1891 2257 2465 2561 2665 2743 3277)
Base 15 - Up to 100000:    70  First 20: (14 341 742 946 1477 1541 1687 1729 1891 1921 2821 3133 3277 4187 6541 6601 7471 8701 8911 9073)
Base 16 - Up to 100000:   200  First 20: (15 51 85 91 255 341 435 451 561 595 645 703 1105 1247 1261 1271 1285 1387 1581 1687)
Base 17 - Up to 100000:    83  First 20: (4 8 9 16 45 91 145 261 781 1111 1228 1305 1729 1885 2149 2821 3991 4005 4033 4187)
Base 18 - Up to 100000:   134  First 20: (25 49 65 85 133 221 323 325 343 425 451 637 931 1105 1225 1369 1387 1649 1729 1921)
Base 19 - Up to 100000:   121  First 20: (6 9 15 18 45 49 153 169 343 561 637 889 905 906 1035 1105 1629 1661 1849 1891)
Base 20 - Up to 100000:   101  First 20: (21 57 133 231 399 561 671 861 889 1281 1653 1729 1891 2059 2413 2501 2761 2821 2947 3059)

Wren

Library: Wren-math

Library: Wren-gmp

Library: Wren-fmt

<lang ecmascript>import "./math" for Int import "./gmp" for Mpz import "./fmt" for Fmt

var isFermatPseudoprime = Fn.new { |x, a|

   if (Int.isPrime(x)) return false
   x = Mpz.from(x)
   a = Mpz.from(a)
   return a.modPow(x-1, x) == 1

}

for (limit in [12000, 25000, 50000, 100000]) {

   for (a in 1..20) {
       var count = 0
       var first20 = List.filled(20, 0)
       for (x in 2..limit) {
           if (isFermatPseudoprime.call(x, a)) {
               if (count < 20) first20[count] = x
               count = count + 1
           }
       }
       Fmt.print("Base $2d - Up to $d: $5d  First 20: $n", a, limit, count, first20)
   }
   System.print()

}</lang>

Output:

Base  1 - Up to 12000: 10561  First 20: [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32]
Base  2 - Up to 12000:    25  First 20: [341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321]
Base  3 - Up to 12000:    25  First 20: [91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601]
Base  4 - Up to 12000:    50  First 20: [15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071]
Base  5 - Up to 12000:    22  First 20: [4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881]
Base  6 - Up to 12000:    31  First 20: [35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729, 2465, 2701, 2821, 3421, 3565, 3589, 3913, 4123, 4495, 5713]
Base  7 - Up to 12000:    21  First 20: [6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321, 10225, 10585, 10621, 11041]
Base  8 - Up to 12000:    76  First 20: [9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949]
Base  9 - Up to 12000:    55  First 20: [4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288]
Base 10 - Up to 12000:    35  First 20: [9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187]
Base 11 - Up to 12000:    30  First 20: [10, 15, 70, 133, 190, 259, 305, 481, 645, 703, 793, 1105, 1330, 1729, 2047, 2257, 2465, 2821, 4577, 4921]
Base 12 - Up to 12000:    35  First 20: [65, 91, 133, 143, 145, 247, 377, 385, 703, 1045, 1099, 1105, 1649, 1729, 1885, 1891, 2041, 2233, 2465, 2701]
Base 13 - Up to 12000:    31  First 20: [4, 6, 12, 21, 85, 105, 231, 244, 276, 357, 427, 561, 1099, 1785, 1891, 2465, 2806, 3605, 5028, 5149]
Base 14 - Up to 12000:    33  First 20: [15, 39, 65, 195, 481, 561, 781, 793, 841, 985, 1105, 1111, 1541, 1891, 2257, 2465, 2561, 2665, 2743, 3277]
Base 15 - Up to 12000:    22  First 20: [14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921, 2821, 3133, 3277, 4187, 6541, 6601, 7471, 8701, 8911, 9073]
Base 16 - Up to 12000:    69  First 20: [15, 51, 85, 91, 255, 341, 435, 451, 561, 595, 645, 703, 1105, 1247, 1261, 1271, 1285, 1387, 1581, 1687]
Base 17 - Up to 12000:    31  First 20: [4, 8, 9, 16, 45, 91, 145, 261, 781, 1111, 1228, 1305, 1729, 1885, 2149, 2821, 3991, 4005, 4033, 4187]
Base 18 - Up to 12000:    46  First 20: [25, 49, 65, 85, 133, 221, 323, 325, 343, 425, 451, 637, 931, 1105, 1225, 1369, 1387, 1649, 1729, 1921]
Base 19 - Up to 12000:    48  First 20: [6, 9, 15, 18, 45, 49, 153, 169, 343, 561, 637, 889, 905, 906, 1035, 1105, 1629, 1661, 1849, 1891]
Base 20 - Up to 12000:    35  First 20: [21, 57, 133, 231, 399, 561, 671, 861, 889, 1281, 1653, 1729, 1891, 2059, 2413, 2501, 2761, 2821, 2947, 3059]

Base  1 - Up to 25000: 22237  First 20: [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32]
Base  2 - Up to 25000:    38  First 20: [341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321]
Base  3 - Up to 25000:    38  First 20: [91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601]
Base  4 - Up to 25000:    75  First 20: [15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071]
Base  5 - Up to 25000:    35  First 20: [4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881]
Base  6 - Up to 25000:    46  First 20: [35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729, 2465, 2701, 2821, 3421, 3565, 3589, 3913, 4123, 4495, 5713]
Base  7 - Up to 25000:    32  First 20: [6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321, 10225, 10585, 10621, 11041]
Base  8 - Up to 25000:   110  First 20: [9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949]
Base  9 - Up to 25000:    83  First 20: [4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288]
Base 10 - Up to 25000:    53  First 20: [9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187]
Base 11 - Up to 25000:    41  First 20: [10, 15, 70, 133, 190, 259, 305, 481, 645, 703, 793, 1105, 1330, 1729, 2047, 2257, 2465, 2821, 4577, 4921]
Base 12 - Up to 25000:    60  First 20: [65, 91, 133, 143, 145, 247, 377, 385, 703, 1045, 1099, 1105, 1649, 1729, 1885, 1891, 2041, 2233, 2465, 2701]
Base 13 - Up to 25000:    51  First 20: [4, 6, 12, 21, 85, 105, 231, 244, 276, 357, 427, 561, 1099, 1785, 1891, 2465, 2806, 3605, 5028, 5149]
Base 14 - Up to 25000:    51  First 20: [15, 39, 65, 195, 481, 561, 781, 793, 841, 985, 1105, 1111, 1541, 1891, 2257, 2465, 2561, 2665, 2743, 3277]
Base 15 - Up to 25000:    31  First 20: [14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921, 2821, 3133, 3277, 4187, 6541, 6601, 7471, 8701, 8911, 9073]
Base 16 - Up to 25000:   102  First 20: [15, 51, 85, 91, 255, 341, 435, 451, 561, 595, 645, 703, 1105, 1247, 1261, 1271, 1285, 1387, 1581, 1687]
Base 17 - Up to 25000:    44  First 20: [4, 8, 9, 16, 45, 91, 145, 261, 781, 1111, 1228, 1305, 1729, 1885, 2149, 2821, 3991, 4005, 4033, 4187]
Base 18 - Up to 25000:    69  First 20: [25, 49, 65, 85, 133, 221, 323, 325, 343, 425, 451, 637, 931, 1105, 1225, 1369, 1387, 1649, 1729, 1921]
Base 19 - Up to 25000:    70  First 20: [6, 9, 15, 18, 45, 49, 153, 169, 343, 561, 637, 889, 905, 906, 1035, 1105, 1629, 1661, 1849, 1891]
Base 20 - Up to 25000:    49  First 20: [21, 57, 133, 231, 399, 561, 671, 861, 889, 1281, 1653, 1729, 1891, 2059, 2413, 2501, 2761, 2821, 2947, 3059]

Base  1 - Up to 50000: 44866  First 20: [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32]
Base  2 - Up to 50000:    55  First 20: [341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321]
Base  3 - Up to 50000:    53  First 20: [91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601]
Base  4 - Up to 50000:   111  First 20: [15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071]
Base  5 - Up to 50000:    54  First 20: [4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881]
Base  6 - Up to 50000:    74  First 20: [35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729, 2465, 2701, 2821, 3421, 3565, 3589, 3913, 4123, 4495, 5713]
Base  7 - Up to 50000:    49  First 20: [6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321, 10225, 10585, 10621, 11041]
Base  8 - Up to 50000:   150  First 20: [9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949]
Base  9 - Up to 50000:   113  First 20: [4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288]
Base 10 - Up to 50000:    65  First 20: [9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187]
Base 11 - Up to 50000:    61  First 20: [10, 15, 70, 133, 190, 259, 305, 481, 645, 703, 793, 1105, 1330, 1729, 2047, 2257, 2465, 2821, 4577, 4921]
Base 12 - Up to 50000:    91  First 20: [65, 91, 133, 143, 145, 247, 377, 385, 703, 1045, 1099, 1105, 1649, 1729, 1885, 1891, 2041, 2233, 2465, 2701]
Base 13 - Up to 50000:    68  First 20: [4, 6, 12, 21, 85, 105, 231, 244, 276, 357, 427, 561, 1099, 1785, 1891, 2465, 2806, 3605, 5028, 5149]
Base 14 - Up to 50000:    69  First 20: [15, 39, 65, 195, 481, 561, 781, 793, 841, 985, 1105, 1111, 1541, 1891, 2257, 2465, 2561, 2665, 2743, 3277]
Base 15 - Up to 50000:    42  First 20: [14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921, 2821, 3133, 3277, 4187, 6541, 6601, 7471, 8701, 8911, 9073]
Base 16 - Up to 50000:   145  First 20: [15, 51, 85, 91, 255, 341, 435, 451, 561, 595, 645, 703, 1105, 1247, 1261, 1271, 1285, 1387, 1581, 1687]
Base 17 - Up to 50000:    63  First 20: [4, 8, 9, 16, 45, 91, 145, 261, 781, 1111, 1228, 1305, 1729, 1885, 2149, 2821, 3991, 4005, 4033, 4187]
Base 18 - Up to 50000:    98  First 20: [25, 49, 65, 85, 133, 221, 323, 325, 343, 425, 451, 637, 931, 1105, 1225, 1369, 1387, 1649, 1729, 1921]
Base 19 - Up to 50000:    93  First 20: [6, 9, 15, 18, 45, 49, 153, 169, 343, 561, 637, 889, 905, 906, 1035, 1105, 1629, 1661, 1849, 1891]
Base 20 - Up to 50000:    66  First 20: [21, 57, 133, 231, 399, 561, 671, 861, 889, 1281, 1653, 1729, 1891, 2059, 2413, 2501, 2761, 2821, 2947, 3059]

Base  1 - Up to 100000: 90407  First 20: [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32]
Base  2 - Up to 100000:    78  First 20: [341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321]
Base  3 - Up to 100000:    78  First 20: [91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601]
Base  4 - Up to 100000:   153  First 20: [15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071]
Base  5 - Up to 100000:    73  First 20: [4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881]
Base  6 - Up to 100000:   104  First 20: [35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729, 2465, 2701, 2821, 3421, 3565, 3589, 3913, 4123, 4495, 5713]
Base  7 - Up to 100000:    73  First 20: [6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321, 10225, 10585, 10621, 11041]
Base  8 - Up to 100000:   218  First 20: [9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949]
Base  9 - Up to 100000:   164  First 20: [4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288]
Base 10 - Up to 100000:    90  First 20: [9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187]
Base 11 - Up to 100000:    89  First 20: [10, 15, 70, 133, 190, 259, 305, 481, 645, 703, 793, 1105, 1330, 1729, 2047, 2257, 2465, 2821, 4577, 4921]
Base 12 - Up to 100000:   127  First 20: [65, 91, 133, 143, 145, 247, 377, 385, 703, 1045, 1099, 1105, 1649, 1729, 1885, 1891, 2041, 2233, 2465, 2701]
Base 13 - Up to 100000:    91  First 20: [4, 6, 12, 21, 85, 105, 231, 244, 276, 357, 427, 561, 1099, 1785, 1891, 2465, 2806, 3605, 5028, 5149]
Base 14 - Up to 100000:    96  First 20: [15, 39, 65, 195, 481, 561, 781, 793, 841, 985, 1105, 1111, 1541, 1891, 2257, 2465, 2561, 2665, 2743, 3277]
Base 15 - Up to 100000:    70  First 20: [14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921, 2821, 3133, 3277, 4187, 6541, 6601, 7471, 8701, 8911, 9073]
Base 16 - Up to 100000:   200  First 20: [15, 51, 85, 91, 255, 341, 435, 451, 561, 595, 645, 703, 1105, 1247, 1261, 1271, 1285, 1387, 1581, 1687]
Base 17 - Up to 100000:    83  First 20: [4, 8, 9, 16, 45, 91, 145, 261, 781, 1111, 1228, 1305, 1729, 1885, 2149, 2821, 3991, 4005, 4033, 4187]
Base 18 - Up to 100000:   134  First 20: [25, 49, 65, 85, 133, 221, 323, 325, 343, 425, 451, 637, 931, 1105, 1225, 1369, 1387, 1649, 1729, 1921]
Base 19 - Up to 100000:   121  First 20: [6, 9, 15, 18, 45, 49, 153, 169, 343, 561, 637, 889, 905, 906, 1035, 1105, 1629, 1661, 1849, 1891]
Base 20 - Up to 100000:   101  First 20: [21, 57, 133, 231, 399, 561, 671, 861, 889, 1281, 1653, 1729, 1891, 2059, 2413, 2501, 2761, 2821, 2947, 3059]