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

Author

Juri-Stepan Gerasimov, Mar 27 2020

Keywords

• nonn

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

Crossrefs

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

Programs

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

Formula

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