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- #!/usr/bin/ruby
- # Smallest number such that k^n - 1 contains n distinct prime divisors.
- # https://oeis.org/A219019
- # Known terms:
- # 3, 4, 7, 8, 16, 11, 79, 44, 81, 91, 1024, 47, 12769, 389, 256, 413, 46656, 373, 1048576, 1000, 4096, 43541
- # New terms found:
- # 12769, 389, 256, 413, 46656, 373
- # PARI/GP program:
- # a(n) = my(k=2); while (omega(k^n-1) != n, k++); k;
- # Lower-bounds:
- # a(19) > 632814
- # a(23) > 446082
- # a(26) > 100948
- # a(27) > 19141
- # Upper-bounds:
- # a(19) <= 1048576
- # New term:
- # a(19) = 1048576 (confirmed by Jinyuan Wang, Feb 13 2023)
- local Num!VERBOSE = true
- local Num!USE_FACTORDB = true
- func a(n, from=2) {
- for k in (from..Inf) {
- var t = (k**n - 1)
- say "[#{n}] Checking k = #{k}: #{t}"
- if (t.is_omega_prime(n)) {
- return k
- }
- }
- }
- var n = 23
- var from = 446082
- say a(n, from)
- __END__
- a(1) = 3
- a(2) = 4
- a(3) = 7
- a(4) = 8
- a(5) = 16
- a(6) = 11
- a(7) = 79
- a(8) = 44
- a(9) = 81
- a(10) = 91
- a(11) = 1024
- a(12) = 47
- a(13) = 12769
- a(14) = 389
- a(15) = 256
- a(16) = 413
- a(17) = 46656
- a(18) = 373
- a(19) = ?
- a(20) = 1000
- a(21) = 4096
- a(22) = 43541
- a(23) = ?
- a(24) = 563
- a(25) = 4096
- a(26) = ?
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