A359362 a(n) = (A001222(n) + 1) * A056239(n), where A001222 counts prime indices and A056239 adds them up.

0, 2, 4, 6, 6, 9, 8, 12, 12, 12, 10, 16, 12, 15, 15, 20, 14, 20, 16, 20, 18, 18, 18, 25, 18, 21, 24, 24, 20, 24, 22, 30, 21, 24, 21, 30, 24, 27, 24, 30, 26, 28, 28, 28, 28, 30, 30, 36, 24, 28, 27, 32, 32, 35, 24, 35, 30, 33, 34, 35, 36, 36, 32, 42, 27, 32, 38

Offset: 1

Author

Gus Wiseman, Dec 31 2022

Keywords

• nonn

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.

Crossrefs

A055396 gives minimum prime index, maximum A061395.

A112798 list prime indices, length A001222, sum A056239.

Cf. A261079, A316413, A326836, A326837, A326844, A326846, A359358.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[(PrimeOmega[n]+1)*Total[primeMS[n]],{n,30}]

Python

from sympy import primepi, factorint
def A359362(n): return (sum((f:=factorint(n)).values())+1)*sum(primepi(p)*e for p, e in f.items()) # Chai Wah Wu, Jan 01 2023

Formula

a(n) = (k + 1) * m, where m and k are the sum and length of the integer partition with Heinz number n.

a(n) = 2*A304818(n) - A261079(n).