A378387 a(n) is the number of proper divisors d of n such that (-d)^n == -d (mod n).

0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 3, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 0, 3, 4, 1, 0, 2, 4, 1, 1, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0

Offset: 1

Author

Juri-Stepan Gerasimov, Dec 23 2024

Keywords

• nonn

Comments

From Robert Israel, Dec 27 2024: (Start)

If n > 1 is odd, a(n) > 0 as d = 1 works.

a(n) = 1 if n is a prime power (A246655). (End)

Examples

			a(4) = 0 because the proper divisors of 4 are 1, 2 and
(-1)^4 (mod 4) is not congruent to 3 (mod 4);
(-2)^4 (mod 4) is not congruent to 2 (mod 4).
a(5) = 1 because the only proper divisor of 5 is 1 and
(-1)^5 (mod 5) == 4 (mod 5).
		

Links

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

Crossrefs

Cf. A032741, A371883.

Programs

Magma

[#[d: d in [1..n-1] | n mod d eq 0 and (-d)^n mod n eq n-d]: n in [1..100]];

Maple

f:= proc(n) nops(select((t -> (-t)&^n + t mod n = 0), numtheory:-divisors(n) minus {n})) end proc:
map(f, [$1..100]); # Robert Israel, Dec 27 2024

Mathematica

a[n_] := DivisorSum[n, 1 &, # < n && PowerMod[n - #, n, n] == n - # &]; Array[a, 100] (* Amiram Eldar, Dec 23 2024 *)

Extensions

Edited by N. J. A. Sloane, Jan 11 2025