A219019 Smallest number k > 1 such that k^n - 1 contains n distinct prime divisors.
3, 4, 7, 8, 16, 11, 79, 44, 81, 91, 1024, 47, 12769, 389, 256, 413, 46656, 373, 1048576, 1000, 4096, 43541
Offset: 1
Examples
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.
Programs
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Maple
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): else fi:od:od:
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Mathematica
L = {}; Do[n = 1; While[Length[FactorInteger[n^k - 1]] != k, n++]; Print@AppendTo[L, n], {k, 15}] (* Giovanni Resta, Nov 10 2012 *) snk[n_]:=Module[{k=2},While[PrimeNu[k^n-1]!=n,k++];k]; Array[snk,22] (* Harvey P. Dale, Mar 27 2025 *) -
PARI
a(n) = my(k=2); while (omega(k^n-1) != n, k++); k; \\ Daniel Suteu, Jul 10 2022
Extensions
a(13)-a(18), a(20)-a(22) from Daniel Suteu, Jul 10 2022
a(19) from Jinyuan Wang, Feb 13 2023
Comments