A384781 -id:A384781 - cp's OEIS frontend

A384854 The number of divisors d of n such that (-d)^d == d (mod n).

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 5, 1, 1, 1, 2, 2, 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, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Jun 10 2025

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

Cf. A000005, A032741, A065295, A384237, A384781.

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

a:= n-> add(`if`((-d)&^d-d mod n=0, 1, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 10 2025

a(n) = sumdiv(n, d, Mod(-d, n)^d == d); \\ Michel Marcus, Jun 11 2025

a(n) = 1 + number of proper divisors h of n such that (-h)^h = h (mod n).

A385392 The number of divisors d of n such that -(d^d) == d (mod n).

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 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, 5, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Jun 27 2025

Cf. A032741, A065295, A384237, A384781, A384854, A385103.

[1+#[d: d in [1..n-1] | n mod d eq 0 and Modexp(d, d, n) eq (n-d)]: n in [1..100]]; // Juri-Stepan Gerasimov, Jun 28 2025

a:= n-> add(`if`(d&^d+d mod n=0, 1, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 27 2025

a[n_] := DivisorSum[n, 1 &, PowerMod[#, #, n] == n-# &]; Array[a, 100] (* Amiram Eldar, Jun 27 2025 *)

a(n) = sumdiv(n, d, -Mod(d, n)^d == d); \\ Michel Marcus, Jun 27 2025

A385540 Number of values of nonnegative s < n such that s^s == (-s)^s == s (mod n).

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 1, 0, 1, 2, 0, 1, 0, 3, 2, 0, 0, 4, 0, 1, 2, 3, 1, 1, 2, 4, 1, 3, 0, 4, 1, 0, 3, 2, 1, 3, 1, 3, 2, 1, 1, 6, 0, 3, 4, 3, 1, 1, 3, 6, 2, 2, 0, 4, 3, 2, 1, 3, 0, 3, 1, 3, 7, 0, 3, 6, 0, 1, 3, 6, 1, 3, 1, 4, 5, 3, 4, 6, 1, 1, 4, 3, 0, 5, 0, 4, 4, 2, 1, 8, 4, 2, 3, 3, 2, 1, 0, 8, 5, 5

Offset: 1

Juri-Stepan Gerasimov, Jul 02 2025

nonn

Cf. A065295, A384781, A385541.

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

a[n_] := Count[Range[0, n-1], ?(PowerMod[#, #, n] == PowerMod[-#, #, n] == # &)]; Array[a, 100] (* _Amiram Eldar, Jul 03 2025 *)

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

A385103 Number of values of s, 0 < s < n, such that -(s^s) == s (mod n).

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 1, 0, 1, 4, 2, 1, 4, 3, 2, 0, 1, 2, 2, 1, 3, 3, 1, 1, 2, 6, 1, 1, 3, 6, 1, 0, 2, 2, 5, 1, 4, 3, 3, 1, 1, 4, 3, 1, 2, 3, 1, 1, 1, 4, 1, 2, 4, 2, 3, 2, 3, 5, 2, 3, 4, 3, 1, 0, 5, 5, 2, 1, 2, 8, 3, 1, 3, 8, 3, 1, 3, 4, 2, 1, 1, 3, 2, 3, 5, 4, 3, 1, 4, 6, 5, 2, 3, 3, 2, 1, 5, 2, 3, 1

Offset: 1

Juri-Stepan Gerasimov, Jun 17 2025

nonn

Cf. A065295, A151821, A373901, A382752, A384781, A384854.

a:= n-> add(`if`(s&^s+s mod n=0, 1, 0), s=1..n-1):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 20 2025

a(n) = sum(s=1, n-1, -Mod(s, n)^s == s); \\ Michel Marcus, Jun 19 2025

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

Original entry on oeis.org

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

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 *)

A385499 a(n) is the smallest integer k such that A385392(k) = n.

Original entry on oeis.org

1, 2, 6, 42, 70, 870, 44070, 547470, 15410670, 168638470

Offset: 1

Juri-Stepan Gerasimov, Jun 30 2025

Cf. A384237, A384781, A384854, A385100, A385391, A385392.

s[n_] := DivisorSum[n, 1 &, PowerMod[#, #, n] == n - # &]; With[{v = Array[s, 45000]}, TakeWhile[Flatten[FirstPosition[v, #] & /@ Range[Max[v]]], NumberQ]] (* Amiram Eldar, Jul 03 2025 *)

a(8)-a(10) from Amiram Eldar, Jul 03 2025

cp's OEIS Frontend

A384854 The number of divisors d of n such that (-d)^d == d (mod n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

PARI

Formula

A385392 The number of divisors d of n such that -(d^d) == d (mod n).

Views

Author

Keywords

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A385540 Number of values of nonnegative s < n such that s^s == (-s)^s == s (mod n).

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

PARI

A385103 Number of values of s, 0 < s < n, such that -(s^s) == s (mod n).

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

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

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

PARI

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

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions

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

Views

Author