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%I #26 Oct 11 2023 03:54:27
%S 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,
%T 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,
%U 30,1,24,1,32,41,1,19,48,1,18,27,48,1,13,1,38,49
%N Sum of odd proper divisors of n. Sum of the odd divisors of n that are less than n.
%H Antti Karttunen, <a href="/A091570/b091570.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F If n is odd, a(n) = A000593(n) - n; if n is even, a(n) = A000593(n). - _Michel Marcus_, Jan 14 2014
%F G.f.: Sum_{k>=1} (2*k-1) * x^(4*k-2) / (1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Apr 13 2021
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)-1)/4 = 0.1612335167... . - _Amiram Eldar_, Oct 11 2023
%e The sum of odd divisors of 9 that are less than 9 is 1 + 3 = 4.
%t 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 *)
%o (PARI) a(n) = sumdiv(n , d, (d%2) * (d<n) * d); \\ _Michel Marcus_, Jan 14 2014
%Y Cf. A000035, A000593, A013661, A066191.
%Y 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).
%K easy,nonn
%O 1,6
%A _Mohammad K. Azarian_, Mar 04 2004
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