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

Author

Gus Wiseman, Aug 18 2025

Keywords

• nonn
• new

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}]