A355733 Number of multisets that can be obtained by choosing a divisor of each prime index of n.

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 3, 4, 1, 2, 3, 4, 2, 5, 2, 3, 2, 3, 4, 4, 3, 4, 4, 2, 1, 4, 2, 6, 3, 6, 4, 7, 2, 2, 5, 4, 2, 6, 3, 4, 2, 6, 3, 4, 4, 5, 4, 4, 3, 7, 4, 2, 4, 6, 2, 7, 1, 7, 4, 2, 2, 6, 6, 6, 3, 4, 6, 6, 4, 6, 7, 4, 2, 5, 2, 2, 5, 4, 4, 7

Offset

1,3

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.

Links

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

Wikipedia, Axiom of choice.

Example

The a(15) = 4 multisets are: {1,1}, {1,2}, {1,3}, {2,3}.
The a(18) = 3 multisets are: {1,1,1}, {1,1,2}, {1,2,2}.

Mathematica

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

Crossrefs

Counting all choices of divisors gives A355731, firsts A355732.

Positions of first appearances are A355734.

Choosing weakly increasing divisors gives A355735, firsts A355736.

Choosing only prime divisors gives A355744.

The version choosing a divisor of each number from 1 to n is A355747.

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.

A061395 selects the maximum prime index.

A120383 lists numbers divisible by all of their 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, A076610, A289509, A316524, A335433, A335448, A340606, A344616, A355737, A355739, A355740, A355741, A355746.

Sequence in context: A329656 A000374 A355735 * A256757 A333860 A277314

Adjacent sequences: A355730 A355731 A355732 * A355734 A355735 A355736

Keyword

nonn

Author

Gus Wiseman, Jul 16 2022

Status

approved