A346877 Sum of the divisors, except for the largest, of the n-th odd number.

0, 1, 1, 1, 4, 1, 1, 9, 1, 1, 11, 1, 6, 13, 1, 1, 15, 13, 1, 17, 1, 1, 33, 1, 8, 21, 1, 17, 23, 1, 1, 41, 19, 1, 27, 1, 1, 49, 19, 1, 40, 1, 23, 33, 1, 21, 35, 25, 1, 57, 1, 1, 87, 1, 1, 41, 1, 29, 65, 25, 12, 45, 31, 1, 47, 1, 27, 105, 1, 1, 51, 25, 35, 81, 1, 1, 81, 37

Offset

1,5

Comments

Sum of aliquot divisors (or aliquot parts) of the n-th odd number.

a(n) has a symmetric representation.

Links

Table of n, a(n) for n=1..78.

Formula

a(n) = A001065(2*n-1).

a(n) = A057427(n-1) + A346879(n).

G.f.: Sum_{k>=0} (2*k + 1) * x^(3*k + 2) / (1 - x^(2*k + 1)). - Ilya Gutkovskiy, Aug 20 2021

Sum_{k=1..n} a(k) = (Pi^2/8 - 1)*n^2 + O(n*log(n)). - Amiram Eldar, Mar 17 2024

Example

For n = 5 the 5th odd number is 9 and the divisors of 9 are [1, 3, 9] and the sum of the divisors of 9 except for the largest is 1 + 3 = 4, so a(5) = 4.

Mathematica

a[n_] := DivisorSigma[1, 2*n - 1] - 2*n + 1; Array[a, 100] (* Amiram Eldar, Aug 20 2021 *)

Prog

(Python)
from sympy import divisors
def a(n): return sum(divisors(2*n-1)[:-1])
print([a(n) for n in range(1, 79)]) # Michael S. Branicky, Aug 20 2021
(PARI) a(n) = sigma(2*n-1) - (2*n-1); \\ Michel Marcus, Aug 20 2021

Crossrefs

Bisection of A001065.

Partial sums give A347153.

Cf. A000203, A005408, A008438, A057427, A153485, A237593, A245092, A244049, A326123, A346869, A346878, A346879, A347154.

Sequence in context: A331148 A128137 A232530 * A231987 A235214 A208606

Adjacent sequences: A346874 A346875 A346876 * A346878 A346879 A346880

Keyword

nonn,easy

Author

Omar E. Pol, Aug 20 2021

Status

approved