A355741 Number of ways to choose a sequence of prime factors, one 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
• mult

Comments

First differs from A355744 at a(169) = 4, A355744(169) = 3.

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 1131 are {2,6,10}, and the a(1131) = 4 choices are: {2,2,2}, {2,2,5}, {2,3,2}, {2,3,5}.
		

Links

• Wikipedia, Cartesian product.

Crossrefs

Positions of 0's are A299174.

The version for all divisors is A355731, firsts A355732.

Choosing prime-power divisors gives A355742.

Positions of 1's are A355743.

Counting multisets instead of sequences gives A355744.

The weakly increasing case is A355745, all divisors A355735.

A001414 adds up distinct prime factors, counted by A001221.

A003963 multiplies together the prime indices of n.

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

A289509 lists numbers with relatively prime prime indices.

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

Cf. A000720, A061395, A076610, A120383, A335433, A355733, A355737, A355739, A355746, A355747.

Programs

Mathematica

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

Formula

Totally multiplicative with a(prime(k)) = A001221(k).