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

1,2

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.

Links

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

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

Example

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.

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 *)

Prog

(PARI) a(n)=my(k=valuation(n, 2)); sigma(n)\(2^(k+1)-1)*(k+1)
(Haskell)
a264740 = sum . map a000265 . a027750_row'
-- Reinhard Zumkeller, Nov 23 2015

Crossrefs

Cf. A000593, A000265, A000203.

Cf. A027750, A086463.

Sequence in context: A364884 A369825 A266411 * A349370 A137621 A242705

Adjacent sequences: A264737 A264738 A264739 * A264741 A264742 A264743

Keyword

nonn,look,mult

Author

Franklin T. Adams-Watters, Nov 22 2015

Status

approved