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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 1-4 of 4 results.

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}]

A387120 Number of ways to choose a different constant integer partition of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 2, 2, 3, 0, 2, 2, 2, 0, 4, 3, 4, 0, 2, 2, 4, 0, 6, 2, 3, 0, 2, 4, 0, 0, 4, 4, 2, 0, 4, 2, 6, 0, 6, 4, 8, 0, 2, 6, 4, 0, 4, 3, 4, 0, 6, 2, 4, 0, 5, 0, 4, 0, 8, 4, 2, 0, 6, 2, 6, 0, 8, 4, 2, 0, 6, 6, 6, 0, 4, 6, 4, 0, 6, 8, 4, 0, 0, 2, 2, 0, 4, 4, 8
Offset: 1

Views

Author

Gus Wiseman, Aug 26 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.

Examples

			The prime indices of 90 are {1,2,2,3}, with choices:
  ((1),(2),(1,1),(3))
  ((1),(1,1),(2),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(1,1,1))
so a(90) = 4.

Crossrefs

For multiset systems see A355529, set systems A367901.
For not necessarily different choices we have A355731, see A355740.
For divisors instead of constant partitions we have A355739 (also the disjoint case).
For prime factors instead of constant partitions we have A387136.
For all instead of just constant partitions we have A387110, disjoint case A383706.
For initial intervals instead of partitions we have A387111.
For strict instead of constant partitions we have A387115.
Twice partitions of this type are counted by A387179, constant-block case of A296122.
Positions of zero are A387180 (non-choosable), complement A387181 (choosable).
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. A000005, A063834, A261049, A299200, A299201, A335433, A335448, A355741, A357977, A387135.

Programs

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

A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

Original entry on oeis.org

1, 2, 3, 2, 7, 6, 15, 0, 6, 14, 56, 6, 101, 30, 21, 0, 297, 12, 490, 14, 45, 112, 1255, 0, 42, 202, 6, 30, 4565, 42, 6842, 0, 168, 594, 105, 12, 21637, 980, 303, 0, 44583, 90, 63261, 112, 42, 2510, 124754, 0, 210, 84, 891, 202, 329931, 12, 392, 0, 1470, 9130
Offset: 1

Views

Author

Gus Wiseman, Aug 26 2025

Keywords

Examples

			The prime factors of 9 are (3,3), and the a(9) = 6 choices are:
  ((3),(2,1))
  ((3),(1,1,1))
  ((2,1),(3))
  ((2,1),(1,1,1))
  ((1,1,1),(3))
  ((1,1,1),(2,1))

Crossrefs

For prime factors instead of partitions we have A008966, see A355741.
Twice partitions of this type are counted by A296122.
For prime indices instead of factors we have A387110, see A387136.
For strict partitions and prime indices we have A387115.
For constant partitions and prime indices we have A387120.
Positions of zero are A387326, for indices apparently A276079 (complement A276078).
Allowing repeated partitions gives A387327, see A299200, A357977.
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. A063834, A261049, A299201, A335433, A335448, A355739, A383706, A387111, A387135.

Programs

  • Mathematica
    Table[Length[Select[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]],UnsameQ@@#&]],{n,30}]

A387116 Number of ways to choose a constant sequence of integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 5, 1, 2, 0, 7, 0, 11, 0, 0, 1, 15, 0, 22, 0, 0, 0, 30, 0, 3, 0, 2, 0, 42, 0, 56, 1, 0, 0, 0, 0, 77, 0, 0, 0, 101, 0, 135, 0, 0, 0, 176, 0, 5, 0, 0, 0, 231, 0, 0, 0, 0, 0, 297, 0, 385, 0, 0, 1, 0, 0, 490, 0, 0, 0, 627, 0, 792, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Aug 21 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.
If n is a prime power prime(x)^y, then a(n) is the number of integer partitions of x; otherwise, a(n) = 0.

Examples

			The a(49) = 5 choices:
  ((4),(4))
  ((3,1),(3,1))
  ((2,2),(2,2))
  ((2,1,1),(2,1,1))
  ((1,1,1,1),(1,1,1,1))

Crossrefs

Positions of zeros are A024619, complement A000961.
Twice-partitions of this type are counted by A047968, see also A296122.
For initial intervals instead of partitions we have A055396, see also A387111.
This is the constant case of A299200, see also A357977, A357982.
For disjoint instead of constant we have A383706.
For distinct instead of constant we have A387110.
For divisors instead of partitions we have A387114, see also A355731, A355739.
For strict partitions instead of partitions we have A387117.
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, A276078, A335433, A335448, A355529, A355741, A387115, A387120, A387133, A387136.

Programs

Formula

a(n) = A000041(A297109(n)).
Showing 1-4 of 4 results.

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