A385662 Number of divisors d of n such that d^d == (-d)^d (mod n).

1, 2, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 3, 1, 4, 1, 5, 1, 4, 1, 3, 1, 6, 2, 3, 3, 4, 1, 5, 1, 5, 1, 3, 1, 6, 1, 3, 1, 6, 1, 5, 1, 4, 2, 3, 1, 8, 2, 5, 1, 4, 1, 7, 1, 6, 1, 3, 1, 8, 1, 3, 2, 6, 1, 5, 1, 4, 1, 5, 1, 9, 1, 3, 2, 4, 1, 5, 1, 8, 3, 3, 1, 8, 1, 3, 1, 6, 1, 8, 1, 4, 1, 3, 1, 10, 1, 5, 2, 6

Offset

1,2

Comments

From Robert Israel, Aug 04 2025: (Start)

If n is divisible by 4, a(n) = A000005(n/2).

If n is odd, a(n) is the number of divisors d of n such that n divides d^d.

If n = 2 * m with m odd, a(n) = A000005(m) + a(m). (End)

Links

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

Maple

f:= proc(n) if n::odd then nops(select(d -> d &^ d mod n = 0, numtheory:-divisors(n)))
       elif n mod 4 = 0 then numtheory:-tau(n/2)
       else numtheory:-tau(n/2) + procname(n/2) fi
end proc:
map(f, [$1..100]); # Robert Israel, Aug 04 2025

Mathematica

a[n_] := DivisorSum[n, 1 &, PowerMod[#, #, n] == PowerMod[-#, #, n] &]; Array[a, 100] (* Amiram Eldar, Aug 04 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)]: n in [1..100]];
(PARI) a(n) = sumdiv(n, d, Mod(d, n)^d == Mod(-d, n)^d); \\ Michel Marcus, Aug 04 2025

Crossrefs

Cf. A000005, A384237, A384834, A384854, A385392, A385318.

Sequence in context: A366979 A126131 A343332 * A138012 A072531 A025818

Adjacent sequences: A385659 A385660 A385661 * A385663 A385664 A385665

Keyword

nonn

Author

Juri-Stepan Gerasimov, Aug 03 2025

Status

approved