A016031 -id:A016031 - cp's OEIS frontend

A330196 Number of unlabeled set-systems covering n vertices with no endpoints.

1, 0, 1, 20, 1754

Offset: 0

Gus Wiseman, Dec 05 2019

A set-system is a finite set of finite nonempty sets. An endpoint is a vertex appearing only once (degree 1).

			Non-isomorphic representatives of the a(3) = 20 set-systems:
  {12}{13}{23}
  {1}{23}{123}
  {12}{13}{123}
  {1}{2}{13}{23}
  {1}{2}{3}{123}
  {1}{12}{13}{23}
  {1}{2}{13}{123}
  {1}{12}{13}{123}
  {1}{12}{23}{123}
  {12}{13}{23}{123}
  {1}{2}{3}{12}{13}
  {1}{2}{12}{13}{23}
  {1}{2}{3}{12}{123}
  {1}{2}{12}{13}{123}
  {1}{2}{13}{23}{123}
  {1}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}
  {1}{2}{3}{12}{13}{123}
  {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}{123}

First differences of the non-covering version A330124.
The "multi" version is A302545.
Unlabeled set-systems with no endpoints counted by vertices are A317794.
Unlabeled set-systems with no endpoints counted by weight are A330054.
Unlabeled set-systems counted by vertices are A000612.
Unlabeled set-systems counted by weight are A283877.
Cf. A007716, A016031, A306005, A317795, A320665, A330052, A330053, A330055.

A118016 Integers of the form 2^k/(k-1).

Original entry on oeis.org

4, 4, 8, 64, 8192, 268435456, 576460752303423488, 5316911983139663491615228241121378304, 904625697166532776746648320380374280103671755200316906558262375061821325312

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, May 11 2006

nonn

			k=5: 2^5/(5-1) = 32/4 = 8.
k=17: 2^17/(17-1) = 131072/16 = 8192.

Cf. A016031 (de Bruijn's sequence: 2^(2^(n-1) - n)).

P:=proc(n) local i,j; for i from 2 by 1 to n do j:=2^i/(i-1); if trunc(j)=j then print(j); fi; od; end: P(5000);

f[n_]:=2^n/n*2;Select[Table[f[n],{n,4,6!}],IntegerQ] (* Vladimir Joseph Stephan Orlovsky, Dec 05 2009 *)
Select[Table[2^n/(n-1),{n,2,500}],IntegerQ] (* Harvey P. Dale, Sep 27 2024 *)

a(n) = 4*A016031(n). - Paolo P. Lava, Nov 10 2006

A344779 Number of distinct length-n necklaces on a size-2 alphabet.

Original entry on oeis.org

2, 1, 2, 1, 2, 3, 4, 2, 4, 3, 6, 9, 12, 20, 32, 16, 32, 36, 68, 57, 138, 123, 252, 378, 504, 420, 1296, 1520, 2176, 2816, 4096, 2048, 4096, 3840, 7040, 7408, 14128, 14212, 29224, 29834, 91332, 87175

Offset: 1

Michel Marcus, Aug 18 2021

They are named P(2)n-sequences in Nellore and Ward article.

Abhinav Nellore and Rachel Ward, Arbitrary-length analogs to de Bruijn sequences, arXiv:2108.07759 [math.CO], 2021.

Cf. A016031.

a(2^n) = A016031(n). - Martin Ehrenstein, Aug 24 2021

a(33)-a(42) from Martin Ehrenstein, Aug 25 2021

cp's OEIS Frontend

A330196 Number of unlabeled set-systems covering n vertices with no endpoints.

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A118016 Integers of the form 2^k/(k-1).

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A344779 Number of distinct length-n necklaces on a size-2 alphabet.

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