A299755 -id:A299755 - cp's OEIS frontend

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

1, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 3, 4, 1, 4, 2, 4, 3, 4, 1, 2, 4, 1, 3, 4, 2, 3, 4, 1, 2, 3, 4, 5, 1, 5, 2, 5, 3, 5, 4, 5, 1, 2, 5, 1, 3, 5, 1, 4, 5, 2, 3, 5, 2, 4, 5, 3, 4, 5, 1, 2, 3, 5, 1, 2, 4, 5, 1, 3, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5

Offset: 1

Gus Wiseman, May 11 2021

			The sets are the columns below:
  1 2 1 3 1 2 1 4 1 2 3 1 1 2 1 5 1 2 3 4 1 1 1 2 2 3 1
      2   3 3 2   4 4 4 2 3 3 2   5 5 5 5 2 3 4 3 4 4 2
              3         4 4 4 3           5 5 5 5 5 5 3
                              4                       5
As a tetrangle, the first four triangles are:
  {1}
  {2},{1,2}
  {3},{1,3},{2,3},{1,2,3}
  {4},{1,4},{2,4},{3,4},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}

Wikiversity, Lexicographic and colexicographic order

Triangle lengths are A000079.
Triangle sums are A001793.
Positions of first appearances are A005183.
Set maxima are A070939.
Set lengths are A124736.
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A296774, A299755, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A036043, A049085, A115623, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Mathematica
```
SortBy[Rest[Subsets[Range[5]]],Last]
```

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

Gus Wiseman, Jun 24 2018

nonn

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).

			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).

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

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

Gus Wiseman, Jun 30 2018

nonn

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).

			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.

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

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

Gus Wiseman, Sep 28 2018

nonn

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).

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

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

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

Gus Wiseman, Sep 28 2018

nonn

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).

			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).

Complement of A319827.

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

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

Gus Wiseman, Jun 28 2018

nonn

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).

			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).

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

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

Gus Wiseman, Jun 28 2018

nonn

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.

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

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

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

Gus Wiseman, Sep 30 2019

nonn

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}.

			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}

Wolfram Language Documentation, CoprimeQ

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.

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)&]

cp's OEIS Frontend

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A316094 FDH numbers of strict integer partitions with odd parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A319828 FDH numbers of strict integer partitions of even numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A316268 FDH numbers of connected strict integer partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327905 FDH numbers of pairwise coprime sets.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica