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A182598 Number of prime factors of form cn+1 for numbers 6^n+1.

%I A182598 #12 Jun 02 2025 03:14:45
%S A182598 1,2,1,2,3,2,2,1,2,1,2,3,2,2,3,2,4,2,2,1,3,2,2,2,3,3,4,3,6,1,3,4,2,5,
%T A182598 5,3,2,5,4,3,4,1,2,2,4,1,5,3,3,6,3,4,5,4,4,3,2,1,3,2,1,3,3,3,8,4,4,2,
%U A182598 4,3,1,5,3,5,4,1,7,5,3,3,3,4,5,3,4,7,2,2,7,5,3,2,4,5,2,3,2,4,6
%N A182598 Number of prime factors of form cn+1 for numbers 6^n+1.
%C A182598 Repeated prime factors are counted.
%H A182598 S. Mustonen, <a href="http://www.survo.fi/papers/MustonenPrimes.pdf">On prime factors of numbers m^n+-1</a>
%H A182598 Seppo Mustonen, <a href="/A182590/a182590.pdf">On prime factors of numbers m^n+-1</a> [Local copy]
%e A182598 For n=6, 6^n-1=46655=5*7*31*43 has three prime factors of form, namely 7=n+1, 31=5n+1, 43=7n+1. Thus a(6)=3.
%t A182598 m = 6; n = 2; nmax = 100;
%t A182598 While[n <= nmax, {l = FactorInteger[m^n + 1]; s = 0;
%t A182598      For[i = 1, i <= Length[l],
%t A182598       i++, {p = l[[i, 1]];
%t A182598        If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
%t A182598      a[n] = s;} n++;];
%t A182598 Table[a[n], {n, 2, nmax}]
%t A182598 Table[{p, e}=Transpose[FactorInteger[6^n+1]]; Sum[If[Mod[p[[i]], n] == 1, e[[i]], 0], {i, Length[p]}], {n, 2, 50}]
%K A182598 nonn
%O A182598 2,2
%A A182598 _Seppo Mustonen_, Nov 24 2010