A355731 Number of ways to choose a sequence of divisors, one of each element of the multiset of prime indices of n (row n of A112798).

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 2, 4, 3, 4, 1, 2, 4, 4, 2, 6, 2, 3, 2, 4, 4, 8, 3, 4, 4, 2, 1, 4, 2, 6, 4, 6, 4, 8, 2, 2, 6, 4, 2, 8, 3, 4, 2, 9, 4, 4, 4, 5, 8, 4, 3, 8, 4, 2, 4, 6, 2, 12, 1, 8, 4, 2, 2, 6, 6, 6, 4, 4, 6, 8, 4, 6, 8, 4, 2, 16, 2, 2, 6, 4, 4

Offset: 1

Author

Gus Wiseman, Jul 16 2022

Keywords

• nonn

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 a(15) = 4 choices are: (1,1), (1,3), (2,1), (2,3).
The a(18) = 4 choices are: (1,1,1), (1,1,2), (1,2,1), (1,2,2).
		

Links

• Wikipedia, Cartesian product.

Crossrefs

Positions of 1's are A000079.

Dominated by A003963 (cf. A049820), with equality at A003586.

Positions of first appearances are A355732.

Counting distinct sequences after sorting gives A355733, firsts A355734.

Requiring the result to be weakly increasing gives A355735, firsts A355736.

Requiring the result to be relatively prime gives A355737, firsts A355738.

Requiring the choices to be distinct gives A355739, zeros A355740.

For prime divisors A355741, prime-powers A355742, weakly increasing A355745.

Choosing divisors of each of 1..n and resorting gives A355747.

An ordered version (using standard order compositions) is A355748.

A000005 counts divisors.

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.

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.

A340852 lists numbers that can be factored into divisors of bigomega.

Cf. A000720, A061395, A076610, A316524, A326841, A335433, A335448, A340606, A344616, A345957, A355744, A355746.

Programs

Mathematica

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

Formula

a(n) = Product_{k=1..A001222(n)} A000005(A112798(n,k)).