A065295 -id:A065295 - 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

A233521 Number of disjoint subsets s of 0..(n-1) such that, for every x in s, x^x (mod n) is in s.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 2, 4, 3, 5, 1, 7, 2, 5, 7, 7, 3, 9, 2, 10, 8, 7, 3, 13, 5, 10, 5, 13, 3, 15, 4, 11, 9, 10, 9, 15, 2, 7, 12, 19, 6, 20, 4, 12, 15, 7, 4, 22, 11, 16, 12, 15, 2, 16, 14, 18, 10, 9, 1, 30, 7, 8, 22, 19, 16, 21, 4, 17, 9, 23, 4, 27, 5, 10, 19, 14, 14

Offset: 1

T. D. Noe, Feb 19 2014

nonn

This is very loosely based on the work of Kurlberg et al. It appears that a(n) = 1 at only six n: 1, 2, 3, 5, 11, 59.

			The simplest nontrivial case is n = 4. In this case, a(4) = 2 because there are two subsets: {0,1,2} and {3}. Note that 0^0 == 1 (mod 4), 1^1 == 1 (mod 4), 2^2 == 0 (mod 4), and 3^3 == 3 (mod 4).

T. D. Noe, Table of n, a(n) for n = 1..1000
Pär Kurlberg, Florian Luca, and Igor Shparlinski, On the fixed points of the map x -> x^x modulo a prime, arXiv 1402.4464

Cf. A065295, A233518, A233519, A233520.

Table[toDo = Range[0, n-1]; sets = {}; While[Length[toDo] > 0, k = toDo[[1]]; toDo = Rest[toDo]; lst = {k}; While[q = PowerMod[k, k, n]; ! MemberQ[lst, q], AppendTo[lst, q]; toDo = Complement[toDo, {q}]; k = q]; AppendTo[sets, lst]]; Do[int = Intersection[sets[[i]], sets[[j]]]; If[int != {}, sets[[i]] = Union[sets[[i]], sets[[j]]]; sets[[j]] = {}], {i, Length[sets]}, {j, i+1, Length[sets]}]; Length[DeleteCases[sets, {}]], {n, 100}]

A385318 Number of nonnegative s < n such that s^s == (-s)^s (mod n).

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 4, 6, 6, 6, 6, 7, 8, 8, 8, 9, 12, 10, 10, 11, 12, 12, 12, 15, 14, 18, 14, 15, 16, 16, 16, 17, 18, 18, 18, 19, 20, 20, 20, 21, 22, 22, 22, 24, 24, 24, 24, 28, 30, 26, 26, 27, 36, 28, 28, 29, 30, 30, 30, 31, 32, 33, 32, 33, 34, 34, 34, 35

Offset: 1

Juri-Stepan Gerasimov, Jul 31 2025

From Robert Israel, Aug 01 2025: (Start)
a(n) = ceiling(n/2) + the number of odd s < n such that 2 * s^s == 0 (mod n).
If n is divisible by 4, there are no such s, so a(n) = n/2.
If n == 2 (mod 4), then s = n/2 works, so a(n) >= n/2 + 1. (End)

Cf. A065295, A384781, A385103, A386409.

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

f:= proc(n) local s;
  ceil(n/2) + nops(select(s -> 2 * s &^ s mod n = 0, [seq(s, s = 1 .. n-1, 2)]))
end proc:
map(f, [$1..100]); # Robert Israel, Aug 01 2025

a[n_] := Count[Range[0, n-1], ?(PowerMod[#, #, n] == PowerMod[-#, #, n] &)]; Array[a, 100] (* _Amiram Eldar, Jul 31 2025 *)

a(n) = sum(s=0, n-1, Mod(s, n)^s == Mod(-s, n)^s); \\ Michel Marcus, Aug 07 2025

A385100 a(n) is the smallest integer k such that A384854(k) = n.

Original entry on oeis.org

1, 2, 66, 182, 30, 858, 4830, 201630, 1829030, 976430, 24877650, 645314670, 3392218830, 17041181430

Offset: 1

Michel Marcus and Juri-Stepan Gerasimov, Jun 17 2025

Cf. A000005, A065295, A384781, A384854, A385499.

f(n) = sumdiv(n, d, Mod(-d, n)^d == d); \\ A384854
a(n) = my(k=1); while(f(k) != n, k++); k; \\ Michel Marcus, Jun 18 2025

a(8)-a(11) from Michel Marcus, Jun 18 2025

a(12)-a(14) from Jinyuan Wang, Jul 01 2025

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

Original entry on oeis.org

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 *)

A385391 a(n) is the smallest integer k such that A384237(k) = n.

Original entry on oeis.org

1, 2, 6, 12, 66, 30, 210, 390, 1365, 2310, 3990, 10920, 2730, 84630, 53130, 87780, 114114, 760760, 2042040, 1345890, 285285, 1902810, 570570, 1141140, 25571910, 30240210, 2282280, 358888530, 514083570, 413092680, 998887890, 761140380, 1155284130, 3082219140, 8125850460, 11532931410, 17440042620, 8254436190

Offset: 1

Michel Marcus and Juri-Stepan Gerasimov, Jun 27 2025

nonn

a(1) = A002110(0), a(2) = A002110(1), a(3) = A002110(2), a(6) = A002110(3), a(7) = A002110(4), a(10) = A002110(5), ...?
a(33) onward > 10^9. - Michael S. Branicky, Jun 30 2025
a(44) = 11125544430. - Robert G. Wilson v, Jul 13 2025

Cf. A002110, A065295, A384237, A384854, A385100.

f[n_] := 1 + Total[ Boole[ PowerMod[#, #, n] == # & /@ Divisors[n]]]; k = 3; t[] := 0; t[1] = 1; t[2] = 2; While[k < 3000000001, a = f@k; If[ t[a] == 0, t[a] = k]; k +=3]; t /@ Range@ 38 (* _Robert G. Wilson v, Jul 13 2025 *)

f(n) = sumdiv(n, d, Mod(d, n)^d == d); \\ A384237
a(n) = my(k=1); while(f(k)!=n, k++); k;

a(28)-a(32) from Michael S. Branicky, Jun 30 2025

a(33)-a(38) from Robert G. Wilson v, Jul 13 2025

A387400 Number of nonnegative s < n such that s^s == s^n (mod n).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 2, 3, 3, 5, 1, 7, 2, 6, 8, 7, 3, 8, 2, 9, 7, 5, 2, 12, 5, 6, 9, 11, 2, 15, 3, 15, 8, 7, 9, 13, 2, 8, 8, 17, 5, 18, 3, 11, 14, 7, 2, 24, 9, 13, 10, 14, 1, 20, 11, 17, 9, 8, 1, 27, 4, 9, 19, 31, 15, 18, 3, 12, 10, 22, 2, 23, 5, 9, 17, 13, 16, 23, 4, 33, 26, 10, 1, 31, 13, 9, 12, 20, 4, 27

Offset: 1

Juri-Stepan Gerasimov, Aug 28 2025

Cf. A065295, A182816.

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

a(n) = sum(s=0, n-1, Mod(s,n)^n == Mod(s, n)^s); \\ Michel Marcus, Aug 30 2025

cp's OEIS Frontend

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

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A233521 Number of disjoint subsets s of 0..(n-1) such that, for every x in s, x^x (mod n) is in s.

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A385318 Number of nonnegative s < n such that s^s == (-s)^s (mod n).

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A385100 a(n) is the smallest integer k such that A384854(k) = n.

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

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A385391 a(n) is the smallest integer k such that A384237(k) = n.

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A387400 Number of nonnegative s < n such that s^s == s^n (mod n).

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