A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 4, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 4, 4, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 4, 6, 6, 4, 89, 2, 1

Offset

1,5

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. This sequence gives the number of ways to choose a strict partition of each prime index of n.

The indices i, where a(i) = 1, form A003586, and the indices j, where a(j) > 1, form A059485. - Ivan N. Ianakiev, Oct 27 2022

Links

Table of n, a(n) for n=1..81.

Example

The a(121) = 9 twice-partitions are: (5)(5), (5)(41), (5)(32), (41)(5), (41)(41), (41)(32), (32)(5), (32)(41), (32)(32).

Mathematica

Table[Times@@Cases[FactorInteger[n], {p_, k_}:>PartitionsQ[PrimePi[p]]^k], {n, 100}]

Prog

(PARI) f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 1] = f9(primepi(f[k, 1]))); factorback(f); \\ Michel Marcus, Oct 26 2022

Crossrefs

Other multiplicative sequences: A003961, A045966, A064988, A064989, A357980.

The non-strict version is A299200.

A horizontal version is A357978, non-strict A357977.

A000040 lists the primes.

A056239 adds up prime indices, row-sums of A112798.

Cf. A000041, A000720, A003964, A063834, A076610, A215366, A273873, A296150, A299201-A299203, A357975, A357979, A357983.

Cf. A003586, A059485.

Sequence in context: A081327 A363279 A205781 * A280444 A030422 A090001

Adjacent sequences: A357979 A357980 A357981 * A357983 A357984 A357985

Keyword

nonn,mult

Author

Gus Wiseman, Oct 25 2022

Status

approved