A385638 Number of nonnegative s < n such that (-s)^s == -s^s (mod n).

1, 2, 1, 3, 2, 3, 3, 7, 5, 5, 5, 7, 6, 7, 7, 14, 8, 11, 9, 11, 10, 11, 11, 15, 14, 13, 17, 15, 14, 15, 15, 30, 16, 17, 17, 23, 18, 19, 19, 23, 20, 21, 21, 23, 23, 23, 23, 31, 27, 29, 25, 27, 26, 35, 27, 31, 28, 29, 29, 31, 30, 31, 32, 62, 32, 33, 33, 35, 34, 35, 35, 47

Offset: 1

Author

Juri-Stepan Gerasimov, Aug 06 2025

Keywords

• nonn
• changed

Comments

Every odd s < n satisfies the condition. An even s works only when n divides 2*s^s. Thus a(n) = floor(n/2) plus the even s that satisfy this test. For an odd prime p >= 3, no even s works, so a(p) = (p - 1) / 2. With 0^0 = 1, s = 0 works only for n = 1 or 2. - Robert P. P. McKone, Aug 07 2025

Crossrefs

Cf. A065295, A384781, A385103, A385318, A386409.

Programs

Magma

[#[s: s in [0..n-1] | Modexp(-s,s,n) eq -Modexp(s,s,n) mod n]: n in [1..100]];

Mathematica

a[n_] := Count[Range[0, n-1], ?(PowerMod[-#, #, n] == Mod[-PowerMod[#, #, n], n] &)]; Array[a, 100] (* _Amiram Eldar, Aug 07 2025 *)