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A182592 Number of prime factors of form cn+1 for numbers 5^n-1.

%I A182592 #14 Jun 02 2025 03:14:18
%S A182592 1,1,1,2,2,1,1,2,3,1,2,1,3,3,2,2,3,3,3,3,4,2,3,4,3,4,3,3,5,2,3,3,4,6,
%T A182592 3,3,6,3,5,2,6,2,3,4,4,1,2,1,6,5,3,3,7,5,3,2,5,2,7,3,5,6,4,4,7,5,8,6,
%U A182592 8,2,3,3,6,5,5,3,7,3,4,2,6,3,3,3,6,4,4,6,5,3,2,5,4,7,5,3,4,5,7,3,10,4,5,8,6,5,2,4,7,3,6,8,5,10,2,3,6,5,7
%N A182592 Number of prime factors of form cn+1 for numbers 5^n-1.
%H A182592 S. Mustonen, <a href="http://www.survo.fi/papers/MustonenPrimes.pdf">On prime factors of numbers m^n+-1</a>
%H A182592 Seppo Mustonen, <a href="/A182590/a182590.pdf">On prime factors of numbers m^n+-1</a> [Local copy]
%e A182592 For n=10, 5^n-1=9765624=2^3*3*11*71*521 has three prime factors of the form cn+1, namely 11=n+1, 71=7n+1, 521=52n+1. Thus a(10)=3.
%t A182592 m = 5; n = 2; nmax = 120;
%t A182592 While[n <= nmax, {l = FactorInteger[m^n - 1]; s = 0;
%t A182592      For[i = 1, i <= Length[l],
%t A182592       i++, {p = l[[i, 1]];
%t A182592        If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
%t A182592      a[n] = s;} n++;];
%t A182592 Table[a[n], {n, 2, nmax}]
%t A182592 Table[Count[FactorInteger[5^n-1][[All,1]],_?(Mod[#,n]==1&)],{n,2,130}] (* _Harvey P. Dale_, Dec 11 2016 *)
%K A182592 nonn
%O A182592 2,4
%A A182592 _Seppo Mustonen_, Nov 22 2010