A355744 Number of multisets that can be obtained by choosing a prime factor of each prime index of n.

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2

Offset: 1

Author

Gus Wiseman, Jul 18 2022

Keywords

• nonn

Comments

First differs from A355741 at a(169) = 3, A355741(169) = 4.

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 a(169) = 3 multisets are: {2,2}, {2,3}, {3,3}.
The a(507) = 3 multisets are: {2,2,2}, {2,2,3}, {2,3,3}.
		

Links

• Wikipedia, Axiom of choice.

Crossrefs

Choosing from all divisors gives A355733, firsts A355734.

Counting sequences instead of multisets gives A355741.

Choosing weakly increasing sequences of divisors gives A355745.

A001414 adds up distinct prime divisors, counted by A001221.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A324850 lists numbers divisible by the product of their prime indices.

A344606 counts alternating permutations of prime indices.

Cf. A000720, A076610, A120383, A289509, A335433, A340852, A355731, A355735, A355737, A355739, A355740.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Tuples[primeMS/@primeMS[n]]]],{n,100}]