cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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

Views

Author

Michel Marcus and Juri-Stepan Gerasimov, Jun 17 2025

Keywords

Crossrefs

Cf. A000005, A065295, A384781, A384854, A385499.

Programs

  • PARI
    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

Extensions

a(8)-a(11) from Michel Marcus, Jun 18 2025
a(12)-a(14) from Jinyuan Wang, Jul 01 2025

A384781 Number of values of s, 0 < s <= n - 1, such that (-s)^s == s (mod n).

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 1, 0, 1, 4, 0, 1, 1, 3, 3, 0, 0, 4, 0, 1, 2, 3, 1, 1, 3, 6, 1, 3, 1, 6, 1, 0, 3, 2, 2, 3, 3, 3, 2, 1, 1, 6, 0, 3, 5, 3, 1, 1, 3, 8, 2, 2, 2, 4, 3, 2, 1, 5, 0, 3, 3, 3, 7, 0, 5, 6, 0, 1, 3, 8, 1, 3, 3, 8, 5, 3, 4, 6, 1, 1, 4, 3, 0, 5, 2, 4, 6, 2, 4, 10, 5, 2, 3, 3, 2, 1, 4, 8, 5, 5
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 09 2025

Keywords

Crossrefs

Cf. A065295, A151821, A373901 (k such that a(k) = 0), A382752, A384854.

Programs

  • Magma
    [#[s: s in [1..n-1] | Modexp((-s),s,n) eq s]: n in [1..100]];
  • Maple
    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 09 2025
  • Mathematica
    a[n_]:=Length[Select[Range[n-1],PowerMod[-#,#,n]==# &]]; Array[a,100] (* Stefano Spezia, Jun 11 2025 *)
  • PARI
    a(n) = sum(s=1, n-1, Mod(-s, n)^s == s); \\ Michel Marcus, Jun 11 2025

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

Views

Author

Juri-Stepan Gerasimov, Jun 27 2025

Keywords

Crossrefs

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

Programs

  • Magma
    [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
  • Maple
    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
  • Mathematica
    a[n_] := DivisorSum[n, 1 &, PowerMod[#, #, n] == n-# &]; Array[a, 100] (* Amiram Eldar, Jun 27 2025 *)
  • PARI
    a(n) = sumdiv(n, d, -Mod(d, n)^d == d); \\ Michel Marcus, Jun 27 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

Views

Author

Juri-Stepan Gerasimov, Jun 17 2025

Keywords

Crossrefs

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

Programs

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

Views

Author

Juri-Stepan Gerasimov, Jul 08 2025

Keywords

Crossrefs

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

Programs

  • Magma
    [#[d: d in Divisors(n) | Modexp(d, d, n) eq n-d and Modexp(-d, d, n) eq n-d]: n in [1..100]];
  • Mathematica
    a[n_]:=Length[Select[Divisors[n],Mod[-#,n]==PowerMod[-#,#,n]==PowerMod[#,#,n]&]];Array[a,100] (* James C. McMahon, Jul 21 2025 *)
  • PARI
    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

Views

Author

Juri-Stepan Gerasimov, Jul 02 2025

Keywords

Crossrefs

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

Programs

  • Magma
    [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]];
  • Mathematica
    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

Views

Author

Juri-Stepan Gerasimov, Aug 03 2025

Keywords

Comments

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)

Crossrefs

Cf. A000005, A384237, A384834, A384854, A385392, A385318.

Programs

  • Magma
    [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]];
  • Maple
    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
  • Mathematica
    a[n_] := DivisorSum[n, 1 &, PowerMod[#, #, n] == PowerMod[-#, #, n] &]; Array[a, 100] (* Amiram Eldar, Aug 04 2025 *)
  • PARI
    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

Views

Author

Juri-Stepan Gerasimov, Aug 08 2025

Keywords

Crossrefs

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

Programs

  • Magma
    [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]];
  • Mathematica
    a[n_] := DivisorSum[n, 1 &, PowerMod[-#, #, n] == Mod[-PowerMod[#, #, n], n] &]; Array[a, 100] (* Amiram Eldar, Aug 09 2025 *)
  • PARI
    a(n) = sumdiv(n, d, Mod(-d, n)^d == - Mod(d, n)^d); \\ Michel Marcus, Aug 09 2025

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

Original entry on oeis.org

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

Views

Author

Michel Marcus and Juri-Stepan Gerasimov, Jun 27 2025

Keywords

Comments

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

Crossrefs

Cf. A002110, A065295, A384237, A384854, A385100.

Programs

Extensions

a(28)-a(32) from Michael S. Branicky, Jun 30 2025
a(33)-a(38) from Robert G. Wilson v, Jul 13 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

Views

Author

Juri-Stepan Gerasimov, Jun 30 2025

Keywords

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

a(8)-a(10) from Amiram Eldar, Jul 03 2025
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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