A386930 Number of divisors d of n such that (-d)^d == -d^d (mod n).

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 4, 4, 2, 5, 2, 4, 4, 3, 2, 5, 3, 3, 4, 4, 2, 5, 2, 5, 4, 3, 4, 7, 2, 3, 4, 5, 2, 5, 2, 4, 6, 3, 2, 6, 3, 5, 4, 4, 2, 7, 4, 5, 4, 3, 2, 6, 2, 3, 6, 6, 4, 5, 2, 4, 4, 5, 2, 9, 2, 3, 6, 4, 4, 5, 2, 6, 5, 3, 2, 6, 4, 3, 4, 5, 2, 8, 4, 4, 4, 3, 4, 7, 2, 5, 6, 7

Offset

1,2

Comments

All odd divisors d are included, so a(n) >= A001227(n), with equality if n is odd. On the other hand, if n > 8 is even, d=2 is not included so a(n) <= A000005(n) - 1. - Robert Israel, Sep 08 2025

Links

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

Maple

f:= proc(n) nops(select(d -> (-d)&^ d + d &^ d mod n = 0, numtheory:-divisors(n))) end proc;
map(f, [$1..100]); # Robert Israel, Sep 08 2025

Mathematica

a[n_] := DivisorSum[n, 1 &, PowerMod[-#, #, n] == Mod[-PowerMod[#, #, n], n] &]; Array[a, 100] (* Amiram Eldar, Aug 09 2025 *)

Prog

(Magma) [1 + #[d: d in [1..n-1] | n mod d eq 0 and Modexp(-d, d, n) eq -Modexp(d, d, n) mod n]: n in [1..100]];
(PARI) a(n) = sumdiv(n, d, Mod(-d, n)^d == - Mod(d, n)^d); \\ Michel Marcus, Aug 09 2025

Crossrefs

Cf. A000005, A001227, A384237, A384834, A384854, A385392, A385638.

Sequence in context: A073093 A326196 A222084 * A327394 A088873 A085082

Adjacent sequences: A386927 A386928 A386929 * A386931 A386932 A386933

Keyword

nonn

Author

Juri-Stepan Gerasimov, Aug 08 2025

Status

approved