A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

1, 2, 2, 4, 3, 4, 3, 8, 4, 6, 5, 8, 7, 6, 6, 16, 11, 8, 13, 12, 6, 10, 19, 16, 9, 14, 8, 12, 29, 12, 37, 32, 10, 22, 9, 16, 47, 26, 14, 24, 61, 12, 79, 20, 12, 38, 103, 32, 9, 18, 22, 28, 131, 16, 15, 24, 26, 58, 163, 24, 199, 74, 12, 64, 21, 20, 251, 44, 38

Offset

1,2

Comments

In the definition, taking A000009(k) instead of prime(A000009(k)) gives A357982.

Links

Table of n, a(n) for n=1..69.

Example

We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(1) * prime(1)^2 * prime(2) = 24.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0, 1, Prime[f[i]]], {i, primeMS[n]}];
Array[mtf[PartitionsQ], 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] = prime(f9(primepi(f[k, 1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

Crossrefs

The non-strict version is A357977.

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

A000040 lists the primes.

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

Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357979, A357983.

Sequence in context: A107331 A283187 A324391 * A087808 A217754 A319397

Adjacent sequences: A357975 A357976 A357977 * A357979 A357980 A357981

Keyword

nonn,mult

Author

Gus Wiseman, Oct 24 2022

Status

approved