A182594 Number of prime factors of form c*n+1 for numbers 7^n-1, counted with multiplicity.

1, 1, 2, 1, 2, 2, 1, 3, 3, 2, 2, 1, 4, 2, 3, 2, 4, 2, 3, 1, 4, 3, 4, 3, 3, 4, 4, 3, 3, 3, 2, 4, 3, 3, 3, 3, 4, 4, 5, 3, 4, 2, 4, 2, 4, 2, 3, 4, 4, 5, 5, 3, 5, 1, 6, 3, 4, 4, 5, 4, 6, 2, 3, 6, 6, 4, 6, 3, 8, 2, 5, 5, 5, 3, 2, 3, 7, 2, 5, 6, 7, 7, 3, 3, 9, 5, 4, 3, 6, 5, 5, 4, 3, 3, 5, 3, 11, 4, 6

Offset: 2

Author

Seppo Mustonen, Nov 22 2010

Keywords

• nonn

Examples

			For n=9, 7^n-1 = 40353606 = 2*3^3*19*37*1063 has three prime factors of form, namely 19 = 2n+1, 37 = 4n+1, 1063 = 118n+1. Thus a(9) = 3.
		

Links

• Seppo Mustonen, On prime factors of numbers m^n+-1
• Seppo Mustonen, On prime factors of numbers m^n+-1 [Local copy]

Programs

Maple

f:= proc(n) local F;
  F:= select(t -> t[1] mod n = 1, ifactors(7^n-1)[2]);
  convert(F[..,2],`+`)
end proc:
map(f, [$2..100]); # Robert Israel, Apr 29 2025

Mathematica

m = 7; n = 2; nmax = 80;
While[n <= nmax, {l = FactorInteger[m^n - 1]; s = 0;
     For[i = 1, i <= Length[l],
      i++, {p = l[[i, 1]];
       If[IntegerQ[(p - 1)/n] == True, s = s + l[[i, 2]]];}];
     a[n] = s;} n++;];
Table[a[n], {n, 2, nmax}]

Extensions

Definition clarified and more terms from Robert Israel, Apr 29 2025