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A064132
Number of divisors of 5^n + 1 that are relatively prime to 5^m + 1 for all 0 < m < n.
6
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)
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
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