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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.

User: Thierry Banel

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Thierry Banel has authored 2 sequences.

A373624 Number of distinct subsets of Z/nZ generated by powers.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 6, 5, 8, 7, 8, 5, 12, 7, 10, 12, 14, 6, 14, 7, 17, 15, 10, 5, 24, 11, 14, 15, 22, 7, 24, 9, 24, 15, 12, 20, 30, 10, 14, 21, 35, 9, 30, 9, 26, 30, 10, 5, 42, 15, 22, 18, 34, 7, 30, 20, 46, 21, 14, 5, 51, 13, 18, 43, 42, 30, 30, 9, 35, 15, 40, 9, 62, 13, 20, 33, 40
Offset: 0

Views

Author

Thierry Banel, Jun 11 2024

Keywords

Comments

Choose p in Z/nZ, then generate the finite subset {1,p,p^2,p^3,p^4,...}. It often happens that two different p give the same subset. Therefore, there may be less distinct subsets than n. a(n) gives the numbers of distinct subsets generated by all p in Z/nZ. Note that the subsets generated by 0, 1, -1 are counted. Those subsets are {1,0}, {1}, {1,-1}.

Examples

			a(7) = 5 because there are 5 distinct power generated subsets of Z/7Z, namely 0^i = {1,0}, 1^i = {1}, 2^i = {1,2,4}, 3^i = {1,3,2,6,4,5}, 6^i = {1,6}. 4^i generates the same subset as 2^i (in a different order, but that is irrelevant). 5^i generate the same subset as 3^i (in a different order).

Programs

  • PARI
    a(n) = #Set(vector(n, i, Set(vector(n, j, Mod(i-1, n)^(j-1))))); \\ Michel Marcus, Jun 12 2024

A166226 Bell number n modulo n.

Original entry on oeis.org

0, 0, 2, 3, 2, 5, 2, 4, 6, 5, 2, 1, 2, 12, 5, 3, 2, 13, 2, 12, 15, 5, 2, 9, 3, 18, 10, 3, 2, 27, 2, 12, 4, 5, 0, 1, 2, 24, 28, 27, 2, 23, 2, 8, 5, 5, 2, 33, 24, 20, 49, 39, 2, 5, 27, 28, 34, 5, 2, 57, 2, 36, 6, 51, 47, 19, 2, 52, 15, 25, 2, 49, 2, 42, 22, 71, 59, 19, 2, 44, 23, 5, 2, 65, 84
Offset: 1

Views

Author

Thierry Banel (tbanel(AT)gmail.com), Oct 09 2009

Keywords

Comments

a(n) = 2 (mod n) when n is prime.

Examples

			a(3)=a(5)=a(7)=a(11)=2.

Links

Crossrefs

See the Bell numbers sequence A000110.

Programs

  • Magma
    [Bell(n) mod n: n in [1..100]]; // Vincenzo Librandi, Feb 03 2016
  • Maple
    seq(combinat:-bell(n) mod n, n=1..100); # Robert Israel, Feb 03 2016
  • Mathematica
    Array[n \[Function] Mod[BellB[n], n], 1000] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010 *)
Table[Mod[BellB[n], n], {n, 1, 100}] (* G. C. Greubel, Feb 02 2016 *)

Formula

a(n) = A000110(n) mod n.
a(p^m) = m+1 (mod p) when p is prime and m >= 1 (see Lemma 3.1 in the Hurst/Schultz reference). - Joerg Arndt, Jun 01 2016

Extensions

More terms from R. J. Mathar and J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

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