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

1,2

Comments

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

Links

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

Example

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

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 *)

Prog

(Python3)
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
(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

Crossrefs

Cf. A000217, A340834.

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

Sequence in context: A107972 A238775 A269525 * A364372 A036252 A103463

Adjacent sequences: A341882 A341883 A341884 * A341886 A341887 A341888

Keyword

nonn,easy

Author

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

Status

approved