A333570 Number of nonnegative values c such that c^n == -c (mod n).

1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 4, 3, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 8, 1, 8, 1, 2, 1, 4, 3, 4, 1, 4, 3, 4, 1, 8, 1, 4, 1, 4, 1, 4, 1, 4, 3, 8, 1, 4, 3, 4, 1, 4, 1, 8, 1, 4, 1, 2, 1, 24, 1, 4, 1, 16, 1, 4, 1, 4, 3, 8, 1, 8, 1, 4, 1, 4, 1, 8, 5, 4, 3, 4, 1, 8, 7, 4, 1, 4, 3

Offset

1,2

Comments

a(n) is the number of nonnegative bases c < n such that c^n + c == 0 (mod n).

a(2^k) = 2 for k > 0.

a(p^m) = 1 for odd prime p with m >= 0.

Let fy(n) = (the number of values b in Z/nZ such that b^y = b)/(the number of values c in Z/nZ such that -c^y = c) for nonnegative y, then:

f0(n) = A000012(n),

f1(n) = A026741(n),

f2(n) = A000012(n),

1 <= f3(n) <= n,

f4(n) = A000012(n), ...,

1 <= fn(n) = A182816(n)/a(n) <= n, where fn(n) = n for odd noncomposite numbers A006005 and Carmichael numbers A002997.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = A182816(n)/r for some odd r.

Prog

(Magma) [#[c: c in [0..n-1] | -c^n mod n eq c]: n in [1..95]];
(PARI) a(n) = sum(c=1, n, Mod(c, n)^n == -c); \\ Michel Marcus, Mar 27 2020

Crossrefs

Cf. A000012, A000079, A000961, A002997, A006005, A026741, A182816.

Sequence in context: A327395 A327404 A360112 * A363320 A351032 A280726

Adjacent sequences: A333567 A333568 A333569 * A333571 A333572 A333573

Keyword

nonn

Author

Juri-Stepan Gerasimov, Mar 27 2020

Status

approved