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%I #30 Feb 13 2023 02:59:53
%S 3,4,7,8,16,11,79,44,81,91,1024,47,12769,389,256,413,46656,373,
%T 1048576,1000,4096,43541
%N Smallest number k > 1 such that k^n - 1 contains n distinct prime divisors.
%C 365000 < a(19) <= 1048576; a(20) = 1000; a(21) = 4096; a(22) = 43541. - _Daniel Suteu_, Jul 10 2022
%e a(3) = 7 is the smallest number of the set {k(i)} = {7, 9, 13, 15, 19, 21, ...} where k(i)^3 - 1 contains 3 distinct prime divisors.
%p with(numtheory) :for n from 1 to 10 do:ii:=0:for k from 1 to 10^10 while(ii=0) do:x:=k^n-1:y:=factorset(x):n1:=nops(y):if n1=n then ii:=1: printf ( "%d %d \n",n,k):
%p else fi:od:od:
%t L = {}; Do[n = 1; While[Length[FactorInteger[n^k - 1]] != k, n++]; Print@AppendTo[L, n], {k, 15}] (* _Giovanni Resta_, Nov 10 2012 *)
%o (PARI) a(n) = my(k=2); while (omega(k^n-1) != n, k++); k; \\ _Daniel Suteu_, Jul 10 2022
%Y Cf. A001221, A359070.
%K nonn,more,hard
%O 1,1
%A _Michel Lagneau_, Nov 09 2012
%E a(13)-a(18), a(20)-a(22) from _Daniel Suteu_, Jul 10 2022
%E a(19) from _Jinyuan Wang_, Feb 13 2023
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