A355742 Number of ways to choose a sequence of prime-power divisors, one of each prime index of n. Product of bigomega over the prime indices of n, with multiplicity.

1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 4, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 3, 0, 2, 0, 2, 0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2

Offset: 1

Author

Gus Wiseman, Jul 20 2022

Keywords

• nonn
• mult

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 49 are {4,4}, and the a(49) = 4 choices are: (2,2), (2,4), (4,2), (4,4).
The prime indices of 777 are {2,4,12}, and the a(777) = 6 choices are: (2,2,2), (2,2,3), (2,2,4), (2,4,2), (2,4,3), (2,4,4).
		

Links

• Wikipedia, Cartesian product.

Crossrefs

The unordered version is A001970, row-sums of A061260.

Positions of 1's are A076610, just primes A355743.

Positions of 0's are A299174.

Allowing all divisors (not just primes) gives A355731, firsts A355732.

Choosing only prime factors (not prime-powers) gives A355741.

Counting multisets of primes gives A355744.

The case of weakly increasing primes A355745, all divisors A355735.

A000688 counts factorizations into prime powers.

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.

Cf. A000720, A120383, A279784, A289509, A324850, A355733, A355739, A355746.

Programs

Mathematica

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

Formula

Totally multiplicative with a(prime(k)) = A001222(k).