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

1,13

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.

Links

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

Wikipedia, Cartesian product.

Formula

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

Example

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}.

Mathematica

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

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.

Sequence in context: A191261 A355745 A348213 * A355744 A179010 A292262

Adjacent sequences: A355738 A355739 A355740 * A355742 A355743 A355744

Keyword

nonn,mult

Author

Gus Wiseman, Jul 18 2022

Status

approved