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

Author

Omar E. Pol, Aug 20 2021

Keywords

• nonn
• easy

Comments

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

a(n) has a symmetric representation.

Examples

			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.
		

Crossrefs

Bisection of A001065.

Partial sums give A347153.

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

Programs

Mathematica

a[n_] := DivisorSigma[1, 2*n - 1] - 2*n + 1; Array[a, 100] (* Amiram Eldar, Aug 20 2021 *)
Total[Most[Divisors[#]]]&/@Range[1,161,2] (* Harvey P. Dale, Sep 29 2024 *)

PARI

a(n) = sigma(2*n-1) - (2*n-1); \\ Michel Marcus, Aug 20 2021

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

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