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 1-3 of 3 results.

A319829 FDH numbers of strict integer partitions of odd numbers.

Original entry on oeis.org

2, 4, 6, 7, 10, 11, 12, 16, 18, 19, 20, 21, 25, 26, 30, 31, 33, 34, 35, 36, 41, 46, 47, 48, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 68, 71, 74, 75, 78, 79, 80, 83, 86, 88, 90, 91, 92, 93, 95, 97, 98, 99, 102, 103, 105, 108, 109, 116, 118, 119, 121, 123, 125
Offset: 1

Views

Author

Gus Wiseman, Sep 28 2018

Keywords

Comments

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1, ..., y_k) is f(y_1) * ... * f(y_k).

Examples

			The sequence of all strict integer partitions of odd numbers begins: (1), (3), (2,1), (5), (4,1), (7), (3,2), (9), (6,1), (11), (4,3), (5,2), (13), (8,1), (4,2,1), (15), (7,2), (10,1), (5,4), (6,3), (17), (12,1), (19), (9,2), (8,3), (21), (6,2,1), (7,4), (5,3,1), (11,2), (14,1), (4,3,2).

Crossrefs

Complement of A319828.
Cf. A050376, A064547, A213925, A299755, A299757, A300061, A300063, A319241, A319242, A319827.

Programs

  • Mathematica
    nn=200;
FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}:>2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Select[Range[nn],OddQ[Total[FDfactor[#]/.FDrules]]&]

A319828 FDH numbers of strict integer partitions of even numbers.

Original entry on oeis.org

1, 3, 5, 8, 9, 13, 14, 15, 17, 22, 23, 24, 27, 28, 29, 32, 37, 38, 39, 40, 42, 43, 44, 45, 49, 50, 51, 59, 62, 64, 65, 66, 67, 69, 70, 72, 73, 76, 77, 81, 82, 84, 85, 87, 89, 94, 96, 100, 101, 104, 106, 107, 110, 111, 112, 113, 114, 115, 117, 120, 122, 124
Offset: 1

Views

Author

Gus Wiseman, Sep 28 2018

Keywords

Comments

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1, ..., y_k) is f(y_1) * ... * f(y_k).

Examples

			The sequence of all strict integer partitions of even numbers begins: (), (2), (4), (3,1), (6), (8), (5,1), (4,2), (10), (7,1), (12), (3,2,1), (6,2), (5,3), (14), (9,1), (16).

Crossrefs

Complement of A319827.
Cf. A050376, A064547, A213925, A299755, A299757, A300061, A300063, A319241, A319242, A319829.

Programs

  • Mathematica
    nn=200;
FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}:>2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Select[Range[nn],EvenQ[Total[FDfactor[#]/.FDrules]]&]

A327905 FDH numbers of pairwise coprime sets.

Original entry on oeis.org

2, 6, 8, 10, 12, 14, 18, 20, 21, 22, 24, 26, 28, 32, 33, 34, 35, 38, 40, 42, 44, 46, 48, 50, 52, 55, 56, 57, 58, 62, 63, 66, 68, 70, 74, 75, 76, 77, 80, 82, 84, 86, 88, 91, 93, 94, 95, 96, 98, 99, 100, 104, 106, 110, 112, 114, 116, 118, 122, 123, 125, 126, 132
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2019

Keywords

Comments

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict partition or finite set {y_1,...,y_k} is f(y_1)*...*f(y_k).
We use the Mathematica function CoprimeQ, meaning a singleton is not coprime unless it is {1}.

Examples

			The sequence of terms together with their corresponding coprime sets begins:
   2: {1}
   6: {1,2}
   8: {1,3}
  10: {1,4}
  12: {2,3}
  14: {1,5}
  18: {1,6}
  20: {3,4}
  21: {2,5}
  22: {1,7}
  24: {1,2,3}
  26: {1,8}
  28: {3,5}
  32: {1,9}
  33: {2,7}
  34: {1,10}
  35: {4,5}
  38: {1,11}
  40: {1,3,4}
  42: {1,2,5}

Links

  • Wolfram Language Documentation, CoprimeQ

Crossrefs

Heinz numbers of pairwise coprime partitions are A302696 (all), A302797 (strict), A302569 (with singletons), and A302798 (strict with singletons).
FDH numbers of relatively prime sets are A319827.
Cf. A050376, A056239, A064547, A213925, A259936, A299755, A299757, A304711, A319826, A326675.

Programs

  • Mathematica
    FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
nn=100;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Select[Range[nn],CoprimeQ@@(FDfactor[#]/.FDrules)&]
Showing 1-3 of 3 results.

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