A357853 Fully multiplicative with a(prime(k)) = A000009(k+1).

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 4, 2, 5, 3, 4, 1, 6, 4, 8, 2, 6, 4, 10, 2, 4, 5, 8, 3, 12, 4, 15, 1, 8, 6, 6, 4, 18, 8, 10, 2, 22, 6, 27, 4, 8, 10, 32, 2, 9, 4, 12, 5, 38, 8, 8, 3, 16, 12, 46, 4, 54, 15, 12, 1, 10, 8, 64, 6, 20, 6, 76, 4, 89, 18, 8, 8, 12, 10

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Example

We have 525 = prime(2)*prime(3)*prime(3)*prime(4) so a(525) = Q(3)*Q(4)*Q(4)*Q(5) = 2*2*2*3 = 24, where Q = A000009.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ptf[f_][n_]:=Product[f[i], {i, primeMS[n]}];
Array[ptf[PartitionsQ[#+1]&], 100]

Prog

(PARI) \\ here b(n) = A000009(n).
b(n)=my(A=O(x*x^n)); polcoef(eta(x^2 + A)/eta(x + A), n)
a(n) = { my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); b(1+primepi(p))^e) } \\ Andrew Howroyd, Nov 10 2025

Crossrefs

Other multiplicative sequences: A003961, A064988, A064989, A357852, A357980.

The non-strict version is A003964.

The unshifted horizontal version is A357978, non-strict A357977.

The unshifted version is A357982.

A000040 lists the primes.

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

Cf. A000009, A000720, A076610, A273873, A296150, A357979.

Sequence in context: A383766 A366605 A133755 * A336470 A351093 A070104

Adjacent sequences: A357850 A357851 A357852 * A357854 A357855 A357856

Keyword

nonn,mult

Author

Gus Wiseman, Oct 28 2022

Status

approved