A387112 -id:A387112 - cp's OEIS frontend

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

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

Gus Wiseman, Aug 18 2025

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.

			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.

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.

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}]

A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172

Offset: 1

Gus Wiseman, Aug 27 2025

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.

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

The complement for all partitions appears to be A276078, counted by A052335.
For all partitions we appear to have A276079, counted by A387134.
For divisors instead of strict partitions we have A355740, counted by A370320.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873, see A387111.
For initial intervals instead of strict partitions we have A387113, counted by A387118.
These are the positions of 0 in A387115.
Partitions of this type are counted by A387137, complement A387178.
The complement is A387177.
The version for constant partitions is A387180, counted by A387329.
The complement for constant partitions is A387181, counted by A387330.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together the prime indices of n.
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. A000720, A261049, A270995, A335433, A335448, A355744, A357978, A357980, A383706, A387110, A387120.

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

A387113 Numbers whose prime indices do not have (strictly) choosable initial intervals.

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Aug 24 2025

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.
We say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1,2,3},{1},{1,3},{2}) is not.
This sequence lists all numbers k such that if the prime indices of k are (x1,x2,...,xz), then the sequence of sets (initial intervals) ({1,...,x1},{1,...,x2},...,{1,...,xz}) is not choosable.

			The prime indices of 18 are {1,2,2}, with initial intervals ({1},{1,2},{1,2}), which have choices (1,1,1), (1,1,2), (1,2,1), (1,2,2), and since none of these are strict, 18 is in the sequence.
The prime indices of 85 are {3,7}, with initial intervals {{1,2,3},{1,2,3,4,5,6,7}}, which are choosable, so 85 is in not the sequence.
The prime indices of 90 are {1,2,2,3}, with initial intervals {{1},{1,2},{1,2},{1,2,3}}, which are not choosable, so 90 is in the sequence.
The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   48: {1,1,1,1,2}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

For partitions instead of initial intervals we have A276079, complement A276078.
For prime factors instead of initial intervals we have A355529, complement A368100.
For divisors instead of initial intervals we have A355740, complement A368110.
These are the positions of 0 in A387111, complement A387134.
The complement is A387112.
Partitions of this type are counted by A387118, complement A238873.
For strict partitions instead of initial intervals we have A387137, complement A387176.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A367902 counts choosable set-systems, complement A367903.
A370582 counts sets with choosable prime factors, complement A370583.
A370585 counts maximal subsets with choosable prime factors.
Cf. A003963, A005703, A052335, A239312, A335433, A335448, A355739, A355747, A370592, A383706, A387110.

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

A387177 Numbers whose prime indices have choosable sets of strict integer partitions. Positions of nonzero terms in A387115.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98

Offset: 1

Gus Wiseman, Aug 29 2025

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 prime indices of 50 are {1,3,3}, and {(1),(3),(2,1)} is a valid choice of distinct strict partitions, so 50 is in the sequence.

The version for all partitions appears to be A276078, counted by A052335.
The complement for all partitions appears to be A276079, counted by A387134.
The complement for divisors is A355740, counted by A370320.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
The version for divisors is A368110, counted by A239312.
The version for initial intervals is A387112, counted by A238873, see A387111.
The complement for initial intervals is A387113, counted by A387118.
These are the positions of nonzero terms in A387115.
The complement is A387176.
Partitions of this type are counted by A387178, complement A387137.
The complement for constant partitions is A387180, counted by A387329, see A387120.
The version for constant partitions is A387181, counted by A387330.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A289509 lists numbers with relatively prime prime indices.
Cf. A000720, A120383, A270995, A299200, A335433, A335448, A357978, A357982, A383706, A387110.

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

A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

Original entry on oeis.org

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204

Offset: 1

Gus Wiseman, Aug 30 2025

First differs from A276079 in having 125.
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.
Also numbers n with at least one prime index k such that the multiplicity of prime(k) in the prime factorization of n exceeds the number of divisors of k.

			The prime indices of 60 are {1,1,2,3}, and we have the following 4 choices of constant partitions:
  ((1),(1),(2),(3))
  ((1),(1),(2),(1,1,1))
  ((1),(1),(1,1),(3))
  ((1),(1),(1,1),(1,1,1))
Since none of these is strict, 60 is in the sequence.
The prime indices of 90 are {1,2,2,3}, and we have the following 4 strict choices:
  ((1),(2),(1,1),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(3))
  ((1),(1,1),(2),(1,1,1))
So 90 is not in the sequence.

For prime factors instead of constant partitions we have A355529, counted by A370593.
For divisors instead of constant partitions we have A355740, counted by A370320.
The complement for prime factors is A368100, counted by A370592.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873.
For initial intervals instead of partitions we have A387113, counted by A387118.
These are the positions of zero in A387120.
For strict instead of constant partitions we have A387176, counted by A387137.
The complement for strict partitions is A387177, counted by A387178.
Twice-partitions of this type are counted by A387179, constant-block case of A296122.
The complement is A387181 (nonzeros of A387120), counted by A387330.
Partitions of this type are counted by A387329.
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. A355739, A367771, A387111, A387115.
Cf. A000005, A052335, A063834, A276079, A299200, A299201, A335433, A335448, A355731, A383706, A387110.

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

cp's OEIS Frontend

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

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A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

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A387113 Numbers whose prime indices do not have (strictly) choosable initial intervals.

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A387177 Numbers whose prime indices have choosable sets of strict integer partitions. Positions of nonzero terms in A387115.

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A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

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