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.

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A368101 Numbers of which there is exactly one way to choose a different prime factor of each prime index.

Original entry on oeis.org

1, 3, 5, 11, 15, 17, 31, 33, 39, 41, 51, 55, 59, 65, 67, 83, 85, 87, 93, 109, 111, 123, 127, 129, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 213, 235, 237, 241, 249, 255, 267, 277, 283, 295, 303, 305, 319, 321, 327, 331, 335, 341, 353, 365, 367, 381
Offset: 1

Views

Author

Gus Wiseman, Dec 12 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The prime indices of 2795 are {3,6,14}, with prime factors {{3},{2,3},{2,7}}, and the only choice with different terms is {3,2,7}, so 2795 is in the sequence.
The terms together with their prime indices of prime indices begin:
    1: {}
    3: {{1}}
    5: {{2}}
   11: {{3}}
   15: {{1},{2}}
   17: {{4}}
   31: {{5}}
   33: {{1},{3}}
   39: {{1},{1,2}}
   41: {{6}}
   51: {{1},{4}}
   55: {{2},{3}}
   59: {{7}}
   65: {{2},{1,2}}
   67: {{8}}
   83: {{9}}
   85: {{2},{4}}
   87: {{1},{1,3}}
   93: {{1},{5}}
  109: {{10}}
  111: {{1},{1,1,2}}

Crossrefs

For no choices we have A355529, odd A355535, binary A367907.
Positions of ones in A367771.
The version for binary indices is A367908, positions of ones in A367905.
For any number of choices we have A368100.
For a unique set instead of sequence we have A370647, counted by A370594.
A058891 counts set-systems, covering A003465, connected A323818.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sort A118914, length A001221, sum A001222.
A355741 chooses a prime factor of each prime index, multisets A355744.
Cf. A007716, A355737, A355739, A355740, A355745, A367904, A367906, A370584.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], Length[Select[Tuples[prix/@prix[#]], UnsameQ@@#&]]==1&]

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 2, 3, 7, 0, 11, 5, 6, 0, 15, 2, 22, 0, 10, 7, 30, 0, 6, 11, 0, 0, 42, 6, 56, 0, 14, 15, 15, 0, 77, 22, 22, 0, 101, 10, 135, 0, 6, 30, 176, 0, 20, 6, 30, 0, 231, 0, 21, 0, 44, 42, 297, 0, 385, 56, 10, 0, 33, 14, 490, 0, 60, 15, 627, 0
Offset: 1

Views

Author

Gus Wiseman, Aug 18 2025

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

Examples

			The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
  ((2),(1,1))
  ((1,1),(2))
The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
  ((2),(3))
  ((2),(2,1))
  ((2),(1,1,1))
  ((1,1),(2,1))
  ((1,1),(1,1,1))

Crossrefs

Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).
Allowing repeated partitions gives A299200, A357977, A357982, A357978.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.
For prime factors instead of partitions see A355741, A355742, A387136.
The disjoint case is A383706.
For initial intervals instead of partitions we have A387111.
The case of strict partitions is A387115.
The case of constant partitions is A387120.
Taking each prime factor (instead of index) gives A387133.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A008284, A063834, A261049, A296122, A299201, A335433, A335448, A355745, A387135.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[IntegerPartitions/@prix[n]],UnsameQ@@#&]],{n,100}]

A387111 Number of ways to choose a sequence of distinct positive integers, one in the initial interval of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 1, 4, 0, 2, 2, 5, 0, 6, 3, 4, 0, 7, 0, 8, 0, 6, 4, 9, 0, 6, 5, 0, 0, 10, 1, 11, 0, 8, 6, 9, 0, 12, 7, 10, 0, 13, 2, 14, 0, 2, 8, 15, 0, 12, 2, 12, 0, 16, 0, 12, 0, 14, 9, 17, 0, 18, 10, 4, 0, 15, 3, 19, 0, 16, 4, 20, 0, 21, 11, 4, 0, 16, 4, 22
Offset: 1

Views

Author

Gus Wiseman, Aug 18 2025

Keywords

Comments

The initial interval of a nonnegative integer x is the set {1,...,x}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

Examples

			The prime indices of 75 are (2,3,3), with initial intervals ({1,2},{1,2,3},{1,2,3}), with choices (1,2,3), (1,3,2), (2,1,3), (2,3,1), so a(75) = 4.

Crossrefs

Allowing repeated partitions gives A003963.
For constant instead of distinct we have A055396.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors we have A355739, zeros A355740, strict case of A355731.
For prime factors we have A355741, prime powers A355742, weakly increasing A355745.
For integer partitions we have A387110.
Positions of nonzero terms are A387112 (choosable).
Positions of 0 are A387134 (non-choosable).
A001414 adds up distinct prime divisors, counted by A001221.
A061395 gives greatest prime index.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A326841, A335433, A335448, A355733, A355737, A355747, A357980, A383706, A387120, A387133, A387135.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[Range/@prix[n]],UnsameQ@@#&]],{n,100}]

A368112 Sorted positions of first appearances in A368109 (number of ways to choose a binary index of each binary index).

Original entry on oeis.org

1, 4, 20, 52, 64, 68, 84, 116, 308, 372, 820, 884, 1088, 1092, 1108, 1140, 1396, 1908, 2868, 2932, 3956, 5184, 5188, 5204, 5236, 5492, 6004, 8052, 13376, 13380, 13396, 13428, 13684, 14196, 16244, 17204, 17268, 18292, 19252, 19316, 20340, 22388, 24436, 30580
Offset: 1

Views

Author

Gus Wiseman, Dec 17 2023

Keywords

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

Examples

			The terms together with the corresponding set-systems begin:
    1: {{1}}
    4: {{1,2}}
   20: {{1,2},{1,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
   68: {{1,2},{1,2,3}}
   84: {{1,2},{1,3},{1,2,3}}
  116: {{1,2},{1,3},{2,3},{1,2,3}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}
  820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
  884: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4}}

Crossrefs

For multisets we have A367915, unsorted A367913, firsts A367912.
Sorted positions of first appearances in A368109.
The unsorted version is A368111.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A326031, A326702, A326753, A355741, A367771, A367905, A367906, A367911, A368184.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Tuples[bpe/@bpe[n]]], {n,1000}];
Select[Range[Length[c]], FreeQ[Take[c,#-1],c[[#]]]&]

A368183 Number of sets that can be obtained by choosing a different binary index of each binary index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 2, 1, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 1, 3, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 2, 2, 1, 3, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 0, 3, 1, 1, 0, 1, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Dec 17 2023

Keywords

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

Examples

			The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,3,2), (2,1,3), both permutations of {1,2,3}, so a(52) = 1.

Crossrefs

For sequences we have A367905, firsts A367910, sorted A367911.
Positions of zeros are A367907.
Without distinctness we have A367912, firsts A367913, sorted A367915.
Positions of positive terms are A367906.
For sequences without distinctness: A368109, firsts A368111, sorted A368112.
Positions of first appearances are A368184, sorted A368185.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A355739, A355741, A355744, A367771.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]],UnsameQ@@#&]]],{n,0,100}]

A368184 Least k such that there are exactly n ways to choose a set consisting of a different binary index of each binary index of k.

Original entry on oeis.org

7, 1, 4, 20, 276, 320, 1088, 65856, 66112, 66624, 263232
Offset: 0

Views

Author

Gus Wiseman, Dec 18 2023

Keywords

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

Examples

			The terms together with the corresponding set-systems begin:
      7: {{1},{2},{1,2}}
      1: {{1}}
      4: {{1,2}}
     20: {{1,2},{1,3}}
    276: {{1,2},{1,3},{1,4}}
    320: {{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
  65856: {{1,2,3},{1,4},{1,5}}
  66112: {{1,2,3},{2,4},{1,5}}
  66624: {{1,2,3},{1,2,4},{1,5}}

Crossrefs

For strict sequences: A367910, firsts of A367905, sorted A367911.
For multisets w/o distinctness: A367913, firsts of A367912, sorted A367915.
For sequences w/o distinctness: A368111, firsts of A368109, sorted A368112.
Positions of first appearances in A368183.
The sorted version is A368185.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A326031, A326702, A326753, A355739, A355741, A367771, A367906, A367907.

Programs

  • Mathematica
    nn=10000;
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
q=Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]]],{n,nn}];
k=Max@@Select[Range[Max@@q], SubsetQ[q,Range[#]]&]
Table[Position[q,n][[1,1]],{n,0,k}]

A370591 Number of minimal subsets of {1..n} such that it is not possible to choose a different prime factor of each element (non-choosable).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 4, 4, 7, 11, 16, 16, 30, 30, 39, 73
Offset: 0

Views

Author

Gus Wiseman, Feb 28 2024

Keywords

Examples

			The a(1) = 1 through a(10) = 16 subsets:
{1}  {1}  {1}  {1}    {1}    {1}      {1}      {1}      {1}      {1}
               {2,4}  {2,4}  {2,4}    {2,4}    {2,4}    {2,4}    {2,4}
                             {2,3,6}  {2,3,6}  {2,8}    {2,8}    {2,8}
                             {3,4,6}  {3,4,6}  {4,8}    {3,9}    {3,9}
                                               {2,3,6}  {4,8}    {4,8}
                                               {3,4,6}  {2,3,6}  {2,3,6}
                                               {3,6,8}  {2,6,9}  {2,6,9}
                                                        {3,4,6}  {3,4,6}
                                                        {3,6,8}  {3,6,8}
                                                        {4,6,9}  {4,6,9}
                                                        {6,8,9}  {6,8,9}
                                                                 {2,5,10}
                                                                 {4,5,10}
                                                                 {5,8,10}
                                                                 {3,5,6,10}
                                                                 {5,6,9,10}

Crossrefs

Minimal case of A370583, complement A370582.
For binary indices instead of factors we have A370642, minima of A370637.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, indices A112798, length A001222.
A355741 counts choices of a prime factor of each prime index.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368098 counts choosable unlabeled multiset partitions, complement A368097.
A368100 ranks choosable multisets, complement A355529.
A368414 counts choosable factorizations, complement A368413.
A370585 counts maximal choosable sets.
A370592 counts choosable partitions, complement A370593.
Cf. A000040, A000720, A045778, A133686, A355739, A355744, A355745, A367771, A370584, A370586, A370587, A370589.

Programs

  • Mathematica
    Table[Length[fasmin[Select[Subsets[Range[n]], Length[Select[Tuples[prix/@#],UnsameQ@@#&]]==0&]]], {n,0,15}]

A370809 Greatest number of multisets that can be obtained by choosing a prime factor of each part of an integer partition of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 4, 4, 6, 4, 6, 6, 6, 6, 8, 6, 8, 8, 9, 8, 10, 9, 12, 10, 12, 12, 12, 12, 16, 13, 16, 16, 18, 16, 20, 18, 20, 20, 24, 20, 24, 24, 24, 26, 30, 26, 30, 30, 32, 32, 36, 32, 36, 36, 40, 38, 42, 40, 45, 44, 48
Offset: 0

Views

Author

Gus Wiseman, Mar 05 2024

Keywords

Examples

			For the partition (10,6,3,2) there are 4 choices: {2,2,2,3}, {2,2,3,3}, {2,2,3,5}, {2,3,3,5} so a(21) >= 4.
For the partitions of 6 we have the following choices:
  (6): {{2},{3}}
  (51): {}
  (42): {{2,2}}
  (411): {}
  (33): {{3,3}}
  (321): {}
  (3111): {}
  (222): {{2,2,2}}
  (2211): {}
  (21111): {}
  (111111): {}
So a(6) = 2.

Crossrefs

For just all divisors (not just prime factors) we have A370808.
The version for factorizations is A370817, for all divisors A370816.
A000041 counts integer partitions, strict A000009.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741, A355744, A355745 choose prime factors of prime indices.
A368413 counts non-choosable factorizations, complement A368414.
A370320 counts non-condensed partitions, ranks A355740.
A370592, A370593, A370594, `A370807 count non-choosable partitions.
Cf. A000792, A048249, A063834, A239312, A319055, A339095, A355529, A355733, A367771, A368100, A370585.

Programs

  • Mathematica
    Table[Max[Length[Union[Sort /@ Tuples[If[#==1,{},First/@FactorInteger[#]]& /@ #]]]&/@IntegerPartitions[n]],{n,0,30}]

Extensions

Terms a(31) onward from Max Alekseyev, Sep 17 2024

A368185 Sorted list of positions of first appearances in A368183 (number of sets that can be obtained by choosing a different binary index of each binary index).

Original entry on oeis.org

1, 4, 7, 20, 276, 320, 1088, 65856, 66112, 66624
Offset: 1

Views

Author

Gus Wiseman, Dec 18 2023

Keywords

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

Examples

			The terms together with the corresponding set-systems begin:
      1: {{1}}
      4: {{1,2}}
      7: {{1},{2},{1,2}}
     20: {{1,2},{1,3}}
    276: {{1,2},{1,3},{1,4}}
    320: {{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
  65856: {{1,2,3},{1,4},{1,5}}
  66112: {{1,2,3},{2,4},{1,5}}
  66624: {{1,2,3},{1,2,4},{1,5}}

Crossrefs

For sequences we have A367911, unsorted A367910, firsts of A367905.
Multisets w/o distinctness: A367915, unsorted A367913, firsts of A367912.
Sequences w/o distinctness: A368112, unsorted A368111, firsts of A368109.
Sorted list of positions of first appearances in A368183.
The unsorted version is A368184.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A326031, A326702, A326753, A355739, A355741, A367771, A367906, A367907.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]]],{n,1000}];
Select[Range[Length[c]], FreeQ[Take[c,#-1],c[[#]]]&]

A370807 Number of integer partitions of n into parts > 1 such that it is not possible to choose a different prime factor of each part.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 1, 4, 4, 8, 9, 15, 17, 25, 30, 43, 54, 72, 87, 115, 139, 181, 224, 283, 342, 429, 519, 647, 779, 967
Offset: 0

Views

Author

Gus Wiseman, Mar 04 2024

Keywords

Examples

			The a(0) = 0 through a(11) = 9 partitions:
  .  .  .  .  (22)  .  (33)   (322)  (44)    (333)   (55)     (443)
                       (42)          (332)   (432)   (82)     (533)
                       (222)         (422)   (522)   (433)    (542)
                                     (2222)  (3222)  (442)    (632)
                                                     (622)    (722)
                                                     (3322)   (3332)
                                                     (4222)   (4322)
                                                     (22222)  (5222)
                                                              (32222)

Crossrefs

These partitions are ranked by the odd terms of A355529, complement A368100.
The version for set-systems is A367903, complement A367902.
The version for factorizations is A368413, complement A368414.
With ones allowed we have A370593, complement A370592.
For a unique choice we have A370594, ranks A370647.
The version for divisors instead of factors is A370804, complement A370805.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts condensed partitions, ranks A368110.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A133686, A355739, A355740, A367771, A367867, A367905, A370583, A370585, A370586, A370636.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],FreeQ[#,1] && Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#],UnsameQ@@#&]]==0&]],{n,0,30}]
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