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

0,1

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

Table of n, a(n) for n=0..100.

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

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 *)

Prog

(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

Crossrefs

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

Cf. A211241.

Sequence in context: A368795 A194564 A284690 * A072865 A364002 A322728

Adjacent sequences: A064129 A064130 A064131 * A064133 A064134 A064135

Keyword

nonn

Author

Robert G. Wilson v, Sep 10 2001

Extensions

More terms from Robert Israel, Jun 25 2018

Incorrect Mma program deleted by Editors, Jul 02 2018

Status

approved