A341885 a(n) is the sum of A000217(p) over the prime factors p of n, counted with multiplicity.

0, 3, 6, 6, 15, 9, 28, 9, 12, 18, 66, 12, 91, 31, 21, 12, 153, 15, 190, 21, 34, 69, 276, 15, 30, 94, 18, 34, 435, 24, 496, 15, 72, 156, 43, 18, 703, 193, 97, 24, 861, 37, 946, 72, 27, 279, 1128, 18, 56, 33, 159, 97, 1431, 21, 81, 37, 196, 438, 1770, 27, 1891, 499, 40, 18, 106, 75, 2278, 159, 282

Offset: 1

Author

J. M. Bergot and Robert Israel, Feb 22 2021

Keywords

• nonn
• easy

Comments

By definition, this sequence is completely additive. - Peter Munn, Aug 14 2022

Examples

			18 = 2*3*3 so a(18) = 2*3/2 + 3*4/2 + 3*4/2 = 15.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A340834.

For other completely additive sequences with primes p mapped to a function of p, see A001414.

Programs

Maple

f:= proc(n) local t; add(t[1]*(t[1]+1)/2*t[2], t = ifactors(n)[2]) end proc:
map(f, [$1..100]);

Mathematica

Prepend[Array[Total@ PolygonalNumber@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &, 68, 2], 0] (* Michael De Vlieger, Feb 22 2021 *)

PARI

a(n) = my(f=factor(n), p); sum(k=1, #f~, p=f[k,1]; f[k,2]*p*(p+1)/2); \\ Michel Marcus, Aug 14 2022

Python

from sympy import factorint
def A341885(n): return sum(k*m*(m+1)//2 for m,k in factorint(n).items()) # Chai Wah Wu, Feb 25 2021