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

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, 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, 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

Offset: 0

Author

Robert G. Wilson v, Sep 10 2001

Keywords

• nonn

Comments

From Robert Israel, Jun 26 2018: (Start)

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.

(End)

Links

• Sam Wagstaff, Cunningham Project, Factorizations of 5^n-1, n odd, n<=375

Crossrefs

Cf. A064131, A064133, A064134, A064135, A064136, A064137.

Cf. A211241.

Programs

Maple

f:= n -> nops(select(t -> andmap(m -> igcd(t,5^m+1)=1,[$1..n-1]), numtheory:-divisors(5^n+1))):
map(f, [$0..100]); # Robert Israel, Jun 25 2018

Mathematica

a[n_] := Count[Divisors[5^n+1], d_ /; AllTrue[5^Range[n-1]+1, CoprimeQ[d, #]&]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 100}] (* Jean-François Alcover, Jun 27 2018 *)

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

Extensions

More terms from Robert Israel, Jun 25 2018

Incorrect Mma program deleted by Editors, Jul 02 2018