Juri-Stepan Gerasimov - cp's OEIS frontend

A386310 Number of divisors d of n such that 2*d^d == 0 (mod n).

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

Offset: 1

Juri-Stepan Gerasimov, Aug 20 2025

Cf. A000005, A385429, A385662, A386930.

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

Table[Length[Select[Divisors[n], PowerMod[#, #, n] == Mod[n - PowerMod[#, #, n], n] &]], {n, 1, 100}] (* Vaclav Kotesovec, Aug 23 2025 *)

a(n) = sumdiv(n, d, 2*Mod(d, n)^d == 0); \\ Michel Marcus, Aug 30 2025

A386284 Smallest k for which A386930(k) = n.

Original entry on oeis.org

1, 2, 4, 8, 18, 45, 36, 90, 72, 108, 144, 315, 216, 540, 576, 432, 648, 1350, 864, 2160, 1296, 1728, 4050, 2700, 2592, 3888, 6912, 11340, 5184, 5400, 7776, 10395, 10368, 13500, 20790, 10800, 23328, 24300, 16200, 31185, 31104, 21600, 27000, 40500, 62208, 56700

Offset: 1

Juri-Stepan Gerasimov, Aug 14 2025

Cf. A385638, A386930.

a386930[n_] := DivisorSum[n, 1 &, PowerMod[-#, #, n] == Mod[-PowerMod[#, #, n], n] &];a[n_]:=Module[{k=0},Until[a386930[k]==n,k++];k];Array[a,46] (* James C. McMahon, Aug 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

A386930 Number of divisors d of n such that (-d)^d == -d^d (mod n).

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Aug 08 2025

Cf. A000005, A384237, A384834, A384854, A385392, A385638.

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

a[n_] := DivisorSum[n, 1 &, PowerMod[-#, #, n] == Mod[-PowerMod[#, #, n], n] &]; Array[a, 100] (* Amiram Eldar, Aug 09 2025 *)

a(n) = sumdiv(n, d, Mod(-d, n)^d == - Mod(d, n)^d); \\ Michel Marcus, Aug 09 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 *)

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

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

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

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

User: Juri-Stepan Gerasimov

A386310 Number of divisors d of n such that 2*d^d == 0 (mod n).

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Magma

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A386284 Smallest k for which A386930(k) = n.

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

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A386930 Number of divisors d of n such that (-d)^d == -d^d (mod n).

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Magma

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

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Magma

Maple

Mathematica

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

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Extensions

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

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