A355745 Number of ways to choose a prime factor of each prime index of n (with multiplicity, in weakly increasing order) such that the result is also weakly increasing.

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

Offset

1,13

Comments

First differs from A355741 and A355744 at n = 35.

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.

Links

Table of n, a(n) for n=1..87.

Wikipedia, Cartesian product.

Example

The prime indices of 1469 are {6,30}, and there are five valid choices: (2,2), (2,3), (2,5), (3,3), (3,5), so a(1469) = 5.

Mathematica

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

Crossrefs

Allowing all divisors gives A355735, firsts A355736, reverse A355749.

Not requiring an increasing sequence gives A355741.

Choosing a multiset instead of sequence gives A355744.

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.

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

A355731 chooses of a divisor of each prime index, firsts A355732.

A355733 chooses a multiset of divisors, firsts A355734.

Cf. A000720, A076610, A335433, A340852, A355737, A355739, A355740, A355742.

Sequence in context: A307837 A123671 A191261 * A348213 A355741 A355744

Adjacent sequences: A355742 A355743 A355744 * A355746 A355747 A355748

Keyword

nonn

Author

Gus Wiseman, Jul 18 2022

Status

approved