A385540 -id:A385540 - cp's OEIS frontend

A385729 Number of nonnegative values s < n such that (-s) == (-s)^s == s^s (mod n).

1, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 3, 3, 1, 0, 1, 2, 2, 1, 3, 3, 1, 1, 1, 4, 1, 1, 2, 4, 1, 0, 2, 2, 4, 1, 2, 3, 3, 1, 1, 4, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 1, 0, 3, 5, 2, 1, 2, 6, 3, 1, 1, 4, 3, 1, 3, 4, 2, 1, 1, 3, 2, 3, 3, 4, 1, 1, 1, 4, 4, 2, 3, 3, 2, 1, 1, 2, 3, 1

Offset: 1

Juri-Stepan Gerasimov, Jul 08 2025

nonn

Cf. A065392, A384781, A385540, A385731.

[1] cat [#[s: s in [0..n-1] | Modexp(s, s, n) eq n-s and Modexp(-s, s, n) eq n-s]: n in [2..100]];

a[n_] := Count[Range[0, n-1], ?(PowerMod[#, #, n] == PowerMod[-#, #, n] == Mod[-#, n] &)]; Array[a, 100] (* _Amiram Eldar, Jul 15 2025 *)

a(n) = sum(s=0, n-1, (-s == Mod(s, n)^s) && (-s == Mod(-s, n)^s)); \\ Michel Marcus, Jul 09 2025

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

Original entry on oeis.org

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

A386410 Smallest k for which A385541(k) = n.

Original entry on oeis.org

1, 2, 66, 30, 1386, 858, 34650, 1287678, 6380946, 11411400, 24877650

Offset: 1

Juri-Stepan Gerasimov, Jul 21 2025

Cf. A385540, A385541.

A386410[n_] := DivisorSum[n, 1 &, PowerMod[#, #, n] == PowerMod[-#, #, n] == Mod[#, n] &]; seq[lim_] := Module[{s = Array[A386410, lim]}, TakeWhile[Flatten[FirstPosition[s, #] & /@ Range[Max[s]]], !MissingQ[#] &]]; seq[10^5] (* Amiram Eldar, Jul 21 2025 *)

a(8)-a(11) from Amiram Eldar, Jul 21 2025

A385541 Number of divisors of n such that d^d == (-d)^d == d (mod n).

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Jul 02 2025

nonn

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

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

a[n_] := DivisorSum[n, 1 &, PowerMod[#, #, n] == PowerMod[-#, #, n] == Mod[#, n] &]; Array[a, 100] (* Amiram Eldar, Jul 03 2025 *)

cp's OEIS Frontend

A385729 Number of nonnegative values s < n such that (-s) == (-s)^s == s^s (mod n).

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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

A386410 Smallest k for which A385541(k) = n.

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Extensions

A385541 Number of divisors of n such that d^d == (-d)^d == d (mod n).

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica