A346870 Sum of all divisors, except the smallest and the largest of every number, of the first n positive even numbers.

0, 2, 7, 13, 20, 35, 44, 58, 78, 99, 112, 147, 162, 189, 230, 260, 279, 333, 354, 403, 456, 495, 520, 595, 637, 682, 747, 810, 841, 948, 981, 1043, 1120, 1177, 1250, 1372, 1411, 1474, 1563, 1668, 1711, 1850, 1895, 1986, 2129, 2204, 2253, 2408, 2480, 2596, 2709, 2814

Offset: 1

Author

Omar E. Pol, Aug 18 2021

Keywords

• nonn
• easy

Comments

Partial sums of the even-indexed terms of Chowla's function A048050.

a(n) has a symmetric representation.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Partial sums of A346880.

Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346869.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1)+numtheory[sigma](2*n)-1-2*n)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 19 2021

Mathematica

s[n_] := DivisorSigma[1, 2*n] - 2*n - 1; Accumulate @ Array[s, 50] (* Amiram Eldar, Aug 19 2021 *)

Python

from sympy import divisors
from itertools import accumulate
def A346880(n): return sum(divisors(2*n)[1:-1])
def aupton(nn): return list(accumulate(A346880(n) for n in range(1, nn+1)))
print(aupton(52)) # Michael S. Branicky, Aug 19 2021

Python

from math import isqrt
def A346870(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-3*((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)-n*(n+2) # Chai Wah Wu, Nov 02 2023

Formula

a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, May 15 2023