A385729 -id:A385729 - cp's OEIS frontend

A386409 Number of nonnegative values s < n such that (-s)^s == -s (mod n).

1, 1, 2, 2, 2, 4, 2, 4, 3, 4, 3, 7, 5, 5, 6, 6, 4, 6, 4, 9, 8, 5, 2, 13, 4, 8, 3, 9, 4, 14, 3, 6, 7, 8, 12, 11, 3, 7, 10, 17, 5, 16, 6, 9, 10, 5, 2, 19, 7, 8, 10, 13, 3, 6, 11, 18, 11, 7, 3, 27, 5, 7, 14, 10, 16, 17, 5, 13, 7, 21, 4, 21, 5, 6, 14, 12, 12, 21, 5, 27, 6, 11, 3, 29, 16, 10, 9, 19, 4, 23

Offset: 1

Juri-Stepan Gerasimov, Jul 20 2025

Cf. A065295, A384781, A385103, A385540, A385729.

[#[s: s in [0..n-1] | Modexp(-s, s, n) eq n-s mod n]: n in [1..90]];

a[n_] := Count[Range[0, n-1], ?(PowerMod[-#, #, n] == n - # &)]; a[1] = 1; Array[a, 100] (* _Amiram Eldar, Jul 21 2025 *)

a(n) = sum(s=0, n-1, Mod(-s, n) == Mod(-s, n)^s); \\ Michel Marcus, Aug 22 2025

a(1) corrected by Andrew Howroyd, Aug 22 2025

A385731 Number of divisors d of n such that (-d) == (-d)^d == d^d (mod n).

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Jul 08 2025

nonn

Cf. A000005, A384237, A384854, A385392, A385540, A385729.

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

a[n_]:=Length[Select[Divisors[n],Mod[-#,n]==PowerMod[-#,#,n]==PowerMod[#,#,n]&]];Array[a,100] (* James C. McMahon, Jul 21 2025 *)

a(n) = sumdiv(n, d, (-d == Mod(d, n)^d) && (-d == Mod(-d, n)^d)); \\ Michel Marcus, Jul 09 2025

A386436 Smallest k for which A385731(k) = n.

Original entry on oeis.org

1, 2, 6, 42, 1770, 47058, 547470, 8648458, 623254170

Offset: 1

Juri-Stepan Gerasimov and Michel Marcus, Jul 21 2025

Cf. A385729, A385731, A386410.

a[n_]:=Module[{k=0},Until[Length[Select[Divisors[k], Mod[-#, k]==PowerMod[-#, #, k]==PowerMod[#, #, k]&]]==n,k++];k];Array[a,7] (* James C. McMahon, Aug 06 2025 *)

a(9) from Michel Marcus, Jul 21 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

A386409 Number of nonnegative values s < n such that (-s)^s == -s (mod n).

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A385731 Number of divisors d of n such that (-d) == (-d)^d == d^d (mod n).

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Magma

Mathematica

PARI

A386436 Smallest k for which A385731(k) = n.

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Mathematica

Extensions

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

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PARI