A091570 Sum of odd proper divisors of n. Sum of the odd divisors of n that are less than n.

0, 1, 1, 1, 1, 4, 1, 1, 4, 6, 1, 4, 1, 8, 9, 1, 1, 13, 1, 6, 11, 12, 1, 4, 6, 14, 13, 8, 1, 24, 1, 1, 15, 18, 13, 13, 1, 20, 17, 6, 1, 32, 1, 12, 33, 24, 1, 4, 8, 31, 21, 14, 1, 40, 17, 8, 23, 30, 1, 24, 1, 32, 41, 1, 19, 48, 1, 18, 27, 48, 1, 13, 1, 38, 49

Offset: 1

Author

Mohammad K. Azarian, Mar 04 2004

Keywords

• easy
• nonn

Examples

			The sum of odd divisors of 9 that are less than 9 is 1 + 3 = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A000035, A000593, A013661, A066191.

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), this sequence (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Programs

Mathematica

f[2, e_] := 1; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n, 0]; Array[a, 75] (* Amiram Eldar, Oct 11 2023 *)

PARI

a(n) = sumdiv(n , d, (d%2) * (dMichel Marcus, Jan 14 2014

Formula

If n is odd, a(n) = A000593(n) - n; if n is even, a(n) = A000593(n). - Michel Marcus, Jan 14 2014

G.f.: Sum_{k>=1} (2*k-1) * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Apr 13 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)-1)/4 = 0.1612335167... . - Amiram Eldar, Oct 11 2023