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.

User: John Machacek

John Machacek's wiki page.

John Machacek has authored 5 sequences.

A302251 The number of nonempty antichains in the lattice of set partitions.

Original entry on oeis.org

1, 2, 9, 346, 79814831
Offset: 1

Views

Author

John Machacek, Apr 04 2018

Keywords

Comments

Computing terms in this sequence is analogous to Dedekind's problem which asks for the number of antichains in the Boolean algebra.
This count excludes the empty antichain consisting of no set partitions.

Examples

			For n = 3 the a(3) = 9 nonempty antichains are:
{1/2/3}
{1/23}
{12/3}
{13/2}
{1/23, 12/3}
{1/23, 13/2}
{12/3, 13/2}
{1/23, 12/3, 13/2}
{123}
Here we have used the usual shorthand notation for set partitions where 1/23 denotes {{1}, {2,3}}.

Links

Crossrefs

Equals A302250 - 1, Cf. A000372, A007153, A003182, A014466.

Programs

  • Sage
    [Posets.SetPartitions(n).antichains().cardinality() - 1 for n in range(4)]
# minus removes the empty antichain

A302250 The number of antichains in the lattice of set partitions of an n-element set.

Original entry on oeis.org

2, 3, 10, 347, 79814832
Offset: 1

Views

Author

John Machacek, Apr 04 2018

Keywords

Comments

Computing terms in this sequence is analogous to Dedekind's problem which asks for the number of antichains in the Boolean algebra.
This count includes the empty antichain consisting of no set partitions.

Examples

			For n = 3 the a(3) = 10 antichains are:
  {}
  {1/2/3}
  {1/23}
  {12/3}
  {13/2}
  {1/23, 12/3}
  {1/23, 13/2}
  {12/3, 13/2}
  {1/23, 12/3, 13/2}
  {123}.
Here we have used the usual shorthand notation for set partitions where 1/23 denotes {{1}, {2,3}}.

Links

Crossrefs

Equals A302251 + 1, Cf. A000372, A007153, A003182, A014466.

Programs

  • Sage
    [Posets.SetPartitions(n).antichains().cardinality() for n in range(4)]

A286499 Primes which divide a term of A073935.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 43, 101, 163, 257, 487, 1459, 14407, 26407, 39367, 62501, 65537, 77659, 1020101, 1336337, 86093443, 242121643, 258280327, 3103616899, 4528177054183, 15258789062501, 411782264189299, 21108889701347407, 953735353027359375062501
Offset: 1

Views

Author

John Machacek, May 27 2017

Keywords

Comments

A prime p is in this sequence if and only if p-1 = Product_{i} (p_i)^(a_i) with p_j - 1 = Product_{j
This sequence contains all Fermat primes (A019434).

Examples

			p = 43 is in the sequence because 43-1 = 42 = 2*3*7, 7-1 = 6 = 2*3, 3-1 = 2.

Links

Crossrefs

Cf. A073935.

Programs

  • Mathematica
    upTo[mx_] := Block[{ric}, ric[n_, p_] := If[n < mx, Block[{m = n p}, If[PrimeQ[n + 1], Sow[n+1]; ric[n (n + 1), n+1]]; If[IntegerExponent[n, p] == 1, While[m < mx, ric[m, p]; m *= p]]]]; Sort[Reap[ric[1, 2]][[2, 1]]]]; upTo[10^20] (* Giovanni Resta, May 27 2017 *)

Extensions

a(20)-a(29) from Giovanni Resta, May 27 2017

A286497 Prime power Giuga numbers: composite numbers n > 1 such that -1/n + sum 1/p^k = 1, where the sum is over the prime powers p^k dividing n.

Original entry on oeis.org

12, 30, 56, 306, 380, 858, 992, 1722, 2552, 2862, 16256, 30704, 66198, 73712, 86142, 249500, 629802, 1703872, 6127552, 16191736, 19127502, 35359900, 67100672, 101999900, 172173762, 182552538, 266677578, 575688042, 1180712682, 2214408306, 6179139056, 17179738112, 21083999500
Offset: 1

Views

Author

John Machacek, May 27 2017

Keywords

Comments

Since Giuga numbers (A007850) must be squarefree, it follows all Giuga numbers are contained in this sequence.
The number 2^k (2^k-1) is in this sequence whenever 2^k-1 is a Mersenne prime (A000668).

Examples

			n = 12 is in the sequence because -1/12 + 1/2 + 1/2^2 + 1/3 = 1.
n = 18 is NOT in the sequence because -1/18 + 1/2 + 1/3 + 1/3^2 != 1.

Links

Crossrefs

Cf. A000668, A007850.

Programs

  • Mathematica
    ok[n_] := Total[n/Flatten@ Table[e[[1]] ^ Range[e[[2]]], {e, FactorInteger@ n}]] - 1 == n; Select[Range[10^5], ok] (* Giovanni Resta, May 27 2017 *)

Extensions

a(20)-a(31) from Giovanni Resta, May 27 2017
a(32)-a(33) from Giovanni Resta, Jun 26 2017

A283423 Prime power pseudoperfect numbers: numbers m > 1 such that 1/m + Sum 1/p^k = 1, where the sum is over the prime powers p^k | m.

Original entry on oeis.org

2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 64, 100, 128, 162, 256, 272, 294, 342, 486, 500, 512, 1024, 1458, 1806, 2048, 2058, 2500, 4096, 4374, 4624, 6498, 8192, 10100, 12500, 13122, 14406, 16384, 23994, 26406, 32768, 34362, 39366, 47058
Offset: 1

Views

Author

John Machacek, May 27 2017

Keywords

Comments

Since primary pseudoperfect numbers (A054377) must be squarefree, it follows that primary pseudoperfect numbers are contained in this sequence.
This sequence contains all powers of 2. With the exception of the powers of 2, every prime power pseudoperfect number is a pseudoperfect number (A005835).
Every number in A073935 is a prime power pseudoperfect number (note: this sequence and A073935 agree for many terms but eventually differ starting at 23994 the 38th term of this sequence).
The number 2^k(2^k+1) is the sequence whenever 2^k+1 is a Fermat prime (A019434).

Examples

			m = 18 is in the sequence because 1/18 + 1/2 + 1/3 + 1/9 = 1.
m = 12 is NOT in the sequence because 1/12 + 1/2 + 1/4 + 1/3 != 1.

Links

Crossrefs

Cf. A005835, A054377, A073935.

Programs

  • Mathematica
    ok[n_] := Total[n/Flatten@ Table[e[[1]] ^ Range[e[[2]]], {e, FactorInteger[n]}]] + 1 == n; Select[ Range[10^5], ok] (* Giovanni Resta, May 27 2017 *)

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

Content is available under The OEIS End-User License Agreement.