|
%I #27 Mar 21 2024 08:36:17
%S 0,1,2,3,7,8,9,18,19,20,31,32,38,51,52,53,68,81,82,99,100,101,134,135,
%T 143,164,165,182,205,206,207,248,267,268,295,296,297,346,365,366,406,
%U 407,430,463,464,485,520,545,546,603,604,605,692,693,694,735,736,765,830,855
%N Sum of all divisors, except the largest of every number, of the first n odd numbers.
%C Sum of all aliquot divisors (or aliquot parts) of the first n odd numbers.
%C Partial sums of the odd-indexed terms of A001065.
%C a(n) has a symmetric representation.
%F a(n) = A001477(n-1) + A346869(n).
%F G.f.: (1/(1 - x)) * Sum_{k>=0} (2*k + 1) * x^(3*k + 2) / (1 - x^(2*k + 1)). - _Ilya Gutkovskiy_, Aug 20 2021
%F a(n) = (Pi^2/8 - 1)*n^2 + O(n*log(n)). - _Amiram Eldar_, Mar 21 2024
%t s[n_] := DivisorSigma[1, 2*n - 1] - 2*n + 1; Accumulate @ Array[s, 100] (* _Amiram Eldar_, Aug 20 2021 *)
%o (Python)
%o from sympy import divisors
%o from itertools import accumulate
%o def A346877(n): return sum(divisors(2*n-1)[:-1])
%o def aupton(nn): return list(accumulate(A346877(n) for n in range(1, nn+1)))
%o print(aupton(60)) # _Michael S. Branicky_, Aug 20 2021
%o (PARI) a(n) = sum(k=1, n, k = 2*k-1; sigma(k)-k); \\ _Michel Marcus_, Aug 20 2021
%Y Partial sums of A346877.
%Y Cf. A000203, A001065, A001477, A005408, A008438, A048050, A153485, A237593, A245092, A244049, A326123, A346869, A346878, A346879, A347154.
%K nonn,easy
%O 1,3
%A _Omar E. Pol_, Aug 20 2021
|
