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A347153
Sum of all divisors, except the largest of every number, of the first n odd numbers.
3
0, 1, 2, 3, 7, 8, 9, 18, 19, 20, 31, 32, 38, 51, 52, 53, 68, 81, 82, 99, 100, 101, 134, 135, 143, 164, 165, 182, 205, 206, 207, 248, 267, 268, 295, 296, 297, 346, 365, 366, 406, 407, 430, 463, 464, 485, 520, 545, 546, 603, 604, 605, 692, 693, 694, 735, 736, 765, 830, 855
OFFSET
1,3
COMMENTS
Sum of all aliquot divisors (or aliquot parts) of the first n odd numbers.
Partial sums of the odd-indexed terms of A001065.
a(n) has a symmetric representation.
FORMULA
a(n) = A001477(n-1) + A346869(n).
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
a(n) = (Pi^2/8 - 1)*n^2 + O(n*log(n)). - Amiram Eldar, Mar 21 2024
MATHEMATICA
s[n_] := DivisorSigma[1, 2*n - 1] - 2*n + 1; Accumulate @ Array[s, 100] (* Amiram Eldar, Aug 20 2021 *)
PROG
(Python)
from sympy import divisors
from itertools import accumulate
def A346877(n): return sum(divisors(2*n-1)[:-1])
def aupton(nn): return list(accumulate(A346877(n) for n in range(1, nn+1)))
print(aupton(60)) # Michael S. Branicky, Aug 20 2021
(PARI) a(n) = sum(k=1, n, k = 2*k-1; sigma(k)-k); \\ Michel Marcus, Aug 20 2021
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 20 2021
STATUS
approved