cp's OEIS Frontend

A064132 Number of divisors of 5^n + 1 that are relatively prime to 5^m + 1 for all 0 < m < n.

%I A064132 #24 Jan 18 2019 13:56:37
%S A064132 2,4,2,2,2,2,2,4,4,2,4,8,2,4,2,4,4,4,4,8,4,8,4,4,2,4,4,8,2,4,4,8,8,4,
%T A064132 16,4,8,8,4,4,4,16,4,16,2,2,2,8,4,8,8,16,8,8,2,2,16,4,2,16,2,16,4,16,
%U A064132 8,8,4,2,32,8,4,8,4,8,8,16,8,4,16,16,8,8,16,8,8,16,8,8,16,8,8,4,4,8,16,8,8,32,16,2,16
%N A064132 Number of divisors of 5^n + 1 that are relatively prime to 5^m + 1 for all 0 < m < n.
%C A064132 From _Robert Israel_, Jun 26 2018: (Start)
%C A064132 a(n) = Product_{j: A211241(j)=2*n} (1 + e_j) where e_j is the Prime(j)-adic valuation of 5^n+1.  In most cases, each e_j = 1 and a(n) is a power of 2, but a(20243) is divisible by 3 since the multiplicative order of 5 mod 40487 is 40486 and 5^20243+1 is divisible by 40487^2.
%C A064132 (End)
%H A064132 Sam Wagstaff, Cunningham Project, <a href="https://homes.cerias.purdue.edu/~ssw/cun/">Factorizations of 5^n-1, n odd, n<=375</a>
%p A064132 f:= n -> nops(select(t -> andmap(m -> igcd(t,5^m+1)=1,[$1..n-1]), numtheory:-divisors(5^n+1))):
%p A064132 map(f, [$0..100]); # _Robert Israel_, Jun 25 2018
%t A064132 a[n_] := Count[Divisors[5^n+1], d_ /; AllTrue[5^Range[n-1]+1, CoprimeQ[d, #]&]];
%t A064132 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 100}] (* _Jean-François Alcover_, Jun 27 2018 *)
%o A064132 (PARI) a(n) = if (n==0, 2, sumdiv(5^n+1, d, vecsum(vector(n-1, k, gcd(d, 5^k+1) == 1)) == n-1)); \\ _Michel Marcus_, Jun 24 2018
%Y A064132 Cf. A064131, A064133, A064134, A064135, A064136, A064137.
%Y A064132 Cf. A211241.
%K A064132 nonn
%O A064132 0,1
%A A064132 _Robert G. Wilson v_, Sep 10 2001
%E A064132 More terms from _Robert Israel_, Jun 25 2018
%E A064132 Incorrect Mma program deleted by Editors, Jul 02 2018