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A091570
Sum of odd proper divisors of n. Sum of the odd divisors of n that are less than n.
14
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
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
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
KEYWORD
easy,nonn
AUTHOR
Mohammad K. Azarian, Mar 04 2004
STATUS
approved