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

Author

Michel Lagneau, Nov 09 2012

Keywords

• nonn
• more
• hard

Comments

365000 < a(19) <= 1048576; a(20) = 1000; a(21) = 4096; a(22) = 43541. - Daniel Suteu, Jul 10 2022

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.
		

Crossrefs

Cf. A001221, A359070.

Programs

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:

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