A384834 -id:A384834 - cp's OEIS frontend

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

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

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

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

cp's OEIS Frontend

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

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A385662 Number of divisors d of n such that d^d == (-d)^d (mod n).

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Magma

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

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Magma

Mathematica

PARI