A386409 -id:A386409 - cp's OEIS frontend

A384834 Number of divisors of n such that (-d)^d == -d (mod n).

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

Offset: 1

Juri-Stepan Gerasimov, Jul 23 2025

a(n) >= 2 for n > 1, as d = 1 and n always work. a(n) = 2 if n is a prime power (A246655). - Robert Israel, Aug 26 2025

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A386409.

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

a:= n-> add(`if`(0=d+(-d)&^d mod n, 1, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 26 2025

a[n_] := DivisorSum[n, 1 &, PowerMod[-#, #, n] == n-# &]; Array[a, 100] (* Amiram Eldar, Jul 24 2025 *)

a(n) = sumdiv(n, d, Mod(-d, n)^d == n-d); \\ Michel Marcus, Jul 26 2025

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

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 *)

A386557 Smallest k for which A384834(k) = n.

Original entry on oeis.org

1, 2, 10, 6, 60, 42, 210, 780, 420, 2730, 5460, 3570, 10920, 30030, 94710, 231420, 510510, 190190, 1504230, 2552550, 285285, 15120105, 1141140, 570570, 60480420, 78768690, 380570190, 577642065, 514083570

Offset: 1

Juri-Stepan Gerasimov, Jul 25 2025

Cf. A386409, A384834.

f[n_] := DivisorSum[n, 1 &, PowerMod[-#, #, n] == n - # &]; With[{t = Array[f, 10^6]}, TakeWhile[FirstPosition[t, #] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]] (* Amiram Eldar, Jul 26 2025 *)

a(25)-a(29) from Amiram Eldar, Jul 26 2025

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A384834 Number of divisors of n such that (-d)^d == -d (mod n).

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

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

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A386557 Smallest k for which A384834(k) = n.

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