A385318 -id:A385318 - cp's OEIS frontend

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

Juri-Stepan Gerasimov, Aug 06 2025

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

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

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

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

A385662 Number of divisors d of n such that d^d == (-d)^d (mod n).

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 3, 1, 4, 1, 5, 1, 4, 1, 3, 1, 6, 2, 3, 3, 4, 1, 5, 1, 5, 1, 3, 1, 6, 1, 3, 1, 6, 1, 5, 1, 4, 2, 3, 1, 8, 2, 5, 1, 4, 1, 7, 1, 6, 1, 3, 1, 8, 1, 3, 2, 6, 1, 5, 1, 4, 1, 5, 1, 9, 1, 3, 2, 4, 1, 5, 1, 8, 3, 3, 1, 8, 1, 3, 1, 6, 1, 8, 1, 4, 1, 3, 1, 10, 1, 5, 2, 6

Offset: 1

Juri-Stepan Gerasimov, Aug 03 2025

From Robert Israel, Aug 04 2025: (Start)
If n is divisible by 4, a(n) = A000005(n/2).
If n is odd, a(n) is the number of divisors d of n such that n divides d^d.
If n = 2 * m with m odd, a(n) = A000005(m) + a(m). (End)

Cf. A000005, A384237, A384834, A384854, A385392, A385318.

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

f:= proc(n) if n::odd then nops(select(d -> d &^ d mod n = 0, numtheory:-divisors(n)))
       elif n mod 4 = 0 then numtheory:-tau(n/2)
       else numtheory:-tau(n/2) + procname(n/2) fi
end proc:
map(f, [$1..100]); # Robert Israel, Aug 04 2025

a[n_] := DivisorSum[n, 1 &, PowerMod[#, #, n] == PowerMod[-#, #, n] &]; Array[a, 100] (* Amiram Eldar, Aug 04 2025 *)

a(n) = sumdiv(n, d, Mod(d, n)^d == Mod(-d, n)^d); \\ Michel Marcus, Aug 04 2025

A386872 Smallest k for which A385662(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 6, 12, 18, 24, 54, 48, 72, 96, 270, 120, 450, 384, 288, 240, 2310, 360, 1890, 480, 1152, 3150, 4050, 720, 2592, 6930, 1800, 1920, 17010, 1440

Offset: 1

Juri-Stepan Gerasimov, Aug 06 2025

Cf. A385729, A385731, A386410, A385318, A385662.

f(n) = sumdiv(n, d, Mod(d, n)^d == Mod(-d, n)^d); \\ A385662
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Aug 06 2025

cp's OEIS Frontend

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

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Magma

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A385662 Number of divisors d of n such that d^d == (-d)^d (mod n).

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Magma

Maple

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PARI

A386872 Smallest k for which A385662(k) = n, or -1 if no such k exists.

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PARI