A384834 Number of divisors of n such that (-d)^d == -d (mod n).

1, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 2, 3, 2, 3, 3, 3, 2, 6, 2, 3, 3, 3, 2, 2, 2, 3, 3, 4, 2, 3, 3, 3, 3, 3, 2, 5, 2, 3, 3, 2, 4, 5, 2, 3, 3, 5, 2, 4, 2, 3, 3, 3, 2, 5, 2, 3, 2, 3, 2, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3

Offset: 1

Author

Juri-Stepan Gerasimov, Jul 23 2025

Keywords

• nonn
• changed

Comments

a(n) >= 2 for n > 1, as d = 1 and n always work. a(n) = 2 if n is a prime power (A246655). - Robert Israel, Aug 26 2025

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A386409.

Programs

Magma

[1 + #[d: d in Divisors(n) | Modexp(-d,d,n) eq n-d mod n]: n in [1..100]];

Maple

a:= n-> add(`if`(0=d+(-d)&^d mod n, 1, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 26 2025

Mathematica

a[n_] := DivisorSum[n, 1 &, PowerMod[-#, #, n] == n-# &]; Array[a, 100] (* Amiram Eldar, Jul 24 2025 *)

PARI

a(n) = sumdiv(n, d, Mod(-d, n)^d == n-d); \\ Michel Marcus, Jul 26 2025