A347153 Sum of all divisors, except the largest of every number, of the first n odd numbers.

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.

Links

Table of n, a(n) for n=1..60.

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

Crossrefs

Partial sums of A346877.

Cf. A000203, A001065, A001477, A005408, A008438, A048050, A153485, A237593, A245092, A244049, A326123, A346869, A346878, A346879, A347154.

Sequence in context: A123644 A246440 A332699 * A105266 A246448 A054996

Adjacent sequences: A347150 A347151 A347152 * A347154 A347155 A347156

Keyword

nonn,easy

Author

Omar E. Pol, Aug 20 2021

Status

approved