A264740 Sum of odd parts of divisors of n.

1, 2, 4, 3, 6, 8, 8, 4, 13, 12, 12, 12, 14, 16, 24, 5, 18, 26, 20, 18, 32, 24, 24, 16, 31, 28, 40, 24, 30, 48, 32, 6, 48, 36, 48, 39, 38, 40, 56, 24, 42, 64, 44, 36, 78, 48, 48, 20, 57, 62, 72, 42, 54, 80, 72, 32, 80, 60, 60, 72, 62, 64, 104, 7

Offset: 1

Author

Franklin T. Adams-Watters, Nov 22 2015

Keywords

• nonn
• look
• mult

Comments

It is easy to show that a(n) is odd iff n is a square.

a(n) = sigma(n) for odd n, since any divisor of an odd number is odd.

Inverse Möbius transform of A000265(n). - Wesley Ivan Hurt, Jun 26 2025

Examples

			Divisors of 10 are 1, 2, 5, 10. The odd parts of these are 1, 1, 5, 5, so a(10) = 1+1+5+5 = 12.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000593, A000265, A000203.

Cf. A027750, A086463.

Programs

Haskell

a264740 = sum . map a000265 . a027750_row'
-- Reinhard Zumkeller, Nov 23 2015

Maple

with(numtheory): with(padic): seq(add(d/2^ordp(d,2), d in divisors(n)), n=1..80); # Ridouane Oudra, Oct 30 2023

Mathematica

f[p_, e_] := If[p == 2, e + 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 30 2020 *)

PARI

a(n)=my(k=valuation(n,2));sigma(n)\(2^(k+1)-1)*(k+1)

Formula

Multiplicative with a(2^k) = k + 1, a(p^k) = sigma(p^k) = (p^(k+1)-1) / (p-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Nov 04 2022

a(n) = Sum_{d|n} A000265(d). - Ridouane Oudra, Oct 30 2023