A384237 -id:A384237 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A385731 Number of divisors d of n such that (-d) == (-d)^d == d^d (mod n).

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 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, 4, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Jul 08 2025

nonn

Cf. A000005, A384237, A384854, A385392, A385540, A385729.

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

a[n_]:=Length[Select[Divisors[n],Mod[-#,n]==PowerMod[-#,#,n]==PowerMod[#,#,n]&]];Array[a,100] (* James C. McMahon, Jul 21 2025 *)

a(n) = sumdiv(n, d, (-d == Mod(d, n)^d) && (-d == Mod(-d, n)^d)); \\ Michel Marcus, Jul 09 2025

A385541 Number of divisors of n such that d^d == (-d)^d == d (mod n).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 2, 1, 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, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Jul 02 2025

nonn

Cf. A000005, A384237, A384854, A385392, A385540.

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

a[n_] := DivisorSum[n, 1 &, PowerMod[#, #, n] == PowerMod[-#, #, n] == Mod[#, n] &]; Array[a, 100] (* Amiram Eldar, Jul 03 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

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

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

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

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

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A385731 Number of divisors d of n such that (-d) == (-d)^d == d^d (mod n).

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A385541 Number of divisors of n such that d^d == (-d)^d == d (mod n).

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

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

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

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