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.

Previous Showing 31-40 of 40 results.

A316266 FDH numbers of strict integer partitions with prime parts and prime length.

Original entry on oeis.org

12, 21, 28, 33, 44, 57, 75, 76, 77, 84, 100, 123, 132, 133, 141, 164, 175, 183, 188, 209, 228, 231, 244, 249, 275, 287, 291, 300, 308, 329, 332, 363, 388, 399, 417, 427, 451, 453, 475, 484, 492, 507, 517, 525, 532, 556, 564, 581, 591, 604, 627, 671, 676, 679
Offset: 1

Views

Author

Gus Wiseman, Jun 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

			Sequence of strict integer partitions with prime parts and prime length, preceded by their FDH numbers, begins:
  12: (3,2)
  21: (5,2)
  28: (5,3)
  33: (7,2)
  44: (7,3)
  57: (11,2)
  75: (13,2)
  76: (11,3)
  77: (7,5)
  84: (5,3,2)

Crossrefs

Cf. A000586, A045450, A050376, A064547, A213925, A299755, A299757, A316185, A316265, A316267.

Programs

  • Mathematica
    nn=1000;
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],And[PrimeQ[Length[FDfactor[#]]],And@@PrimeQ/@(FDfactor[#]/.FDrules)]&]

A316267 FDH numbers of strict integer partitions of prime numbers with a prime number of prime parts.

Original entry on oeis.org

12, 21, 57, 123, 249, 417, 532, 699, 867, 1100, 1389, 1463, 1509, 1708, 2049, 2068, 2307, 2324, 2913, 3116, 3147, 3157, 3273, 3325, 3619, 3903, 4227, 4268, 4636, 4821, 5079, 5225, 5324, 5516, 5739, 6308, 6391, 6524, 6621, 6644, 7469, 8092, 8193, 8225, 8457
Offset: 1

Views

Author

Gus Wiseman, Jun 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

			Sequence of strict integer partitions of prime numbers with a prime number of prime parts, preceded by their FDH numbers, begins:
    12: (3,2)
    21: (5,2)
    57: (11,2)
   123: (17,2)
   249: (29,2)
   417: (41,2)
   532: (11,5,3)
   699: (59,2)
   867: (71,2)
  1100: (13,7,3)
  1389: (101,2)
  1463: (11,7,5)
  1509: (107,2)
  1708: (23,5,3)

Crossrefs

Cf. A000586, A045450, A050376, A064547, A213925, A299755, A299757, A316185, A316265, A316266.

Programs

  • Mathematica
    nn=1000;
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],And[PrimeQ[Total[FDfactor[#]/.FDrules]],PrimeQ[Length[FDfactor[#]]],And@@PrimeQ/@(FDfactor[#]/.FDrules)]&]

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]]&]

A316094 FDH numbers of strict integer partitions with odd parts.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 14, 16, 19, 22, 25, 28, 31, 32, 38, 41, 44, 47, 50, 53, 56, 61, 62, 64, 71, 76, 77, 79, 82, 83, 88, 94, 97, 100, 101, 103, 106, 107, 109, 112, 113, 121, 122, 124, 127, 128, 131, 133, 137, 139, 142, 149, 151, 152, 154, 157, 158, 163, 164, 166
Offset: 1

Views

Author

Gus Wiseman, Jun 24 2018

Keywords

Comments

Also numbers n such that A305829(n) is odd.
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

			Sequence of all integer partitions with distinct odd parts begins (), (1), (3), (5), (3,1), (7), (5,1), (9), (11), (7,1), (13), (5,3), (15), (9,1), (11,1), (17), (7,3), (19), (13,1), (21), (5,3,1), (23), (15,1), (9,3), (25), (11,3), (7,5), (27), (17,1), (29), (7,3,1), (19,1), (31).

Crossrefs

Cf. A000700, A050376, A064547, A213925, A258116, A279065, A299755, A299757, A305829.

Programs

  • Mathematica
    nn=100;
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[Times@@(FDfactor[#]/.FDrules)]&]

A316361 FDH numbers of strict integer partitions such that not every distinct subset has a different average.

Original entry on oeis.org

24, 56, 60, 110, 120, 135, 140, 168, 210, 216, 224, 264, 270, 273, 280, 308, 312, 315, 330, 342, 360, 378, 384, 408, 420, 440, 456, 459, 480, 504, 520, 540, 546, 550, 552, 576, 585, 594, 600, 616, 630, 660, 672, 693, 696, 702, 728, 744, 756, 759, 760, 770, 780
Offset: 1

Views

Author

Gus Wiseman, Jun 30 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

			210 is the FDH number of (5,4,2,1), and the subsets {1,5}, and {2,4} have the same average, so 210 belongs to the data.

Crossrefs

Cf. A050376, A064547, A108917, A213925, A275972, A299755, A299757, A301899, A301900, A316271, A316313, A316362.

Programs

  • Mathematica
    nn=1000;
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],!UnsameQ@@Mean/@Union[Subsets[FDfactor[#]/.FDrules]]&]

A319825 LCM of the strict integer partition with FDH number n.

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 5, 3, 6, 4, 7, 6, 8, 5, 4, 9, 10, 6, 11, 12, 10, 7, 12, 6, 13, 8, 6, 15, 14, 4, 15, 9, 14, 10, 20, 6, 16, 11, 8, 12, 17, 10, 18, 21, 12, 12, 19, 18, 20, 13, 10, 24, 21, 6, 28, 15, 22, 14, 22, 12, 23, 15, 30, 9, 8, 14, 24, 30, 12, 20, 25, 6, 26
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

			45 is the FDH number of (6,4), which has LCM 12, so a(45) = 12.

Crossrefs

Cf. A050376, A056239, A064547, A213925, A289508, A289509, A290103, A299755, A299757, A319826.

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];
LCM@@@Table[Reverse[FDfactor[n]/.FDrules],{n,2,nn}]

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]]&]

A316264 FDH numbers of strict integer partitions with odd length and all odd parts.

Original entry on oeis.org

2, 4, 7, 11, 16, 19, 25, 31, 41, 47, 53, 56, 61, 71, 79, 83, 88, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 152, 154, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 200, 211, 223, 224, 227, 229, 233, 239, 241, 248, 251, 257
Offset: 1

Views

Author

Gus Wiseman, Jun 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

			Sequence of all strict odd integer partitions begins (1), (3), (5), (7), (9), (11), (13), (15), (17), (19), (21), (1,3,5), (23), (25), (27), (29), (1,3,7), (31).

Crossrefs

Cf. A000700, A050376, A064547, A213925, A258116, A279065, A299755, A299757, A305829, A316094.

Programs

  • Mathematica
    nn=100;
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],And[OddQ[Length[FDfactor[#]]],OddQ[Times@@(FDfactor[#]/.FDrules)]]&]

A316268 FDH numbers of connected strict integer partitions.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 36, 37, 39, 41, 43, 45, 47, 49, 51, 53, 59, 61, 64, 65, 67, 69, 71, 73, 79, 81, 83, 85, 87, 89, 92, 97, 101, 103, 107, 108, 109, 111, 113, 115, 117, 119, 121, 124, 127, 129, 131, 135, 137, 139, 144
Offset: 1

Views

Author

Gus Wiseman, Jun 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).
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set or strict partition S is said to be connected if G(S) is a connected graph.

Examples

			Sequence of connected strict integer partitions begins (1), (2), (3), (4), (5), (6), (7), (8), (4,2), (9), (10), (11), (12), (13), (6,2).

Crossrefs

Cf. A048143, A050376, A064547, A213925, A299755, A299756, A299757, A304714, A304716, A305078, A305079, A305829, A305831.

Programs

  • Mathematica
    nn=100;
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)]]]]];
FDrules=MapIndexed[(#1->#2[[1]])&,Array[FDfactor,nn,1,Union]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>1]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[nn],Length[csm[primeMS/@(FDfactor[#]/.FDrules)]]==1&]

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)&]
Previous Showing 31-40 of 40 results.

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