A355739 Number of ways to choose a sequence of all different divisors, one of each prime index of n (with multiplicity).

1, 1, 2, 0, 2, 1, 3, 0, 2, 1, 2, 0, 4, 2, 3, 0, 2, 0, 4, 0, 4, 1, 3, 0, 2, 3, 0, 0, 4, 1, 2, 0, 3, 1, 5, 0, 6, 3, 6, 0, 2, 1, 4, 0, 2, 2, 4, 0, 6, 0, 3, 0, 5, 0, 3, 0, 6, 3, 2, 0, 6, 1, 2, 0, 6, 1, 2, 0, 5, 2, 6, 0, 4, 5, 2, 0, 5, 2, 4, 0, 0, 1, 2, 0, 3, 3, 6

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, Cartesian product.

Example

The a(49) = 6 ways are: (1,2), (1,4), (2,1), (2,4), (4,1), (4,2).
The a(182) = 5 ways are: (1,2,3), (1,2,6), (1,4,2), (1,4,3), (1,4,6).
The a(546) = 2 ways are: (1,2,4,3), (1,2,4,6).

Mathematica

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

Crossrefs

This is the strict version of A355731, firsts A355732.

For relatively prime instead of strict we have A355737, firsts A355738.

Positions of 0's are A355740.

A000005 counts divisors.

A001221 counts distinct prime factors, with sum A001414.

A001222 counts prime factors with multiplicity.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798.

A120383 lists numbers divisible by all of their prime indices.

A289508 gives GCD of prime indices, positions of 1's A289509.

Cf. A000720, A076610, A302796, A355535, A355537, A355733, A355735, A355741, A355742, A355744, A355748.

Sequence in context: A061986 A127185 A159780 * A324285 A138036 A354856

Adjacent sequences: A355736 A355737 A355738 * A355740 A355741 A355742

Keyword

nonn

Author

Gus Wiseman, Jul 18 2022

Status

approved