A367870 a(n) = Sum_{d|n, d odd} (n-d).

0, 1, 2, 3, 4, 8, 6, 7, 14, 14, 10, 20, 12, 20, 36, 15, 16, 41, 18, 34, 52, 32, 22, 44, 44, 38, 68, 48, 28, 96, 30, 31, 84, 50, 92, 95, 36, 56, 100, 74, 40, 136, 42, 76, 192, 68, 46, 92, 90, 119, 132, 90, 52, 176, 148, 104, 148, 86, 58, 216, 60, 92, 274, 63, 176, 216

Offset: 1

Author

Wesley Ivan Hurt, Dec 03 2023

Keywords

• nonn
• look

Comments

Total distance from n to each odd divisor of n.

Examples

			a(15) = 36. The total distance from 15 to each of its odd divisors is (15-1) + (15-3) + (15-5) + (15-15) = 36.
		

Links

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

Crossrefs

Cf. A000593, A001227, A094471, A245579.

Programs

Maple

f:= proc(n) local x,d;
  x:= n/2^padic:-ordp(n,2);
  add(n-d, d = numtheory:-divisors(x))
end proc:
map(f, [$1..100]); # Robert Israel, Dec 04 2023

Mathematica

Table[DivisorSum[n, n-# &, OddQ], {n, 100}] (* Paolo Xausa, Mar 05 2024 *)

PARI

a(n) = sumdiv(n, d, if (d%2, n-d)); \\ Michel Marcus, Dec 04 2023

Python

from math import prod
from sympy import factorint
def A367870(n):
    f = factorint(n>>(~n&n-1).bit_length())
    return n*prod(e+1 for e in f.values())-prod((p**(e+1)-1)//(p-1) for p,e in f.items()) # Chai Wah Wu, Dec 31 2023

Formula

a(n) = A245579(n) - A000593(n).

a(n) = n*A001227(n) - A000593(n).