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

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sums of divisors.

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

Example

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

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 *)

Prog

(PARI) a(n) = sumdiv(n , d, (d%2) * (d<n) * d); \\ Michel Marcus, Jan 14 2014

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).

Sequence in context: A222479 A285788 A293434 * A116669 A016523 A026998

Adjacent sequences: A091567 A091568 A091569 * A091571 A091572 A091573

Keyword

easy,nonn

Author

Mohammad K. Azarian, Mar 04 2004

Status

approved