A378387 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

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)

LINKS

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

EXAMPLE

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).

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 *)

PROG

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

CROSSREFS

Cf. A032741, A371883.

Sequence in context: A298941 A317146 A194297 * A100544 A031214 A130654

Adjacent sequences: A378384 A378385 A378386 * A378388 A378389 A378390

KEYWORD

nonn

AUTHOR

Juri-Stepan Gerasimov, Dec 23 2024

EXTENSIONS

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

STATUS

approved