A347154 Sum of all divisors, except the largest of every number, of the first n positive even numbers.

1, 4, 10, 17, 25, 41, 51, 66, 87, 109, 123, 159, 175, 203, 245, 276, 296, 351, 373, 423, 477, 517, 543, 619, 662, 708, 774, 838, 870, 978, 1012, 1075, 1153, 1211, 1285, 1408, 1448, 1512, 1602, 1708, 1752, 1892, 1938, 2030, 2174, 2250, 2300, 2456, 2529, 2646, 2760

Offset: 1

Author

Omar E. Pol, Aug 20 2021

Keywords

• nonn
• easy

Comments

Sum of all aliquot divisors (or aliquot parts) of the first n positive even numbers.

Partial sums of the even-indexed terms of A001065.

a(n) has a symmetric representation.

Links

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

Crossrefs

Partial sums of A346878.

Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346870, A346877, A346880, A347153.

Programs

Mathematica

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

PARI

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

Python

from sympy import divisors
from itertools import accumulate
def A346878(n): return sum(divisors(2*n)[:-1])
def aupton(nn): return list(accumulate(A346878(n) for n in range(1, nn+1)))
print(aupton(51)) # Michael S. Branicky, Aug 20 2021

Python

from math import isqrt
def A347154(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+1) # Chai Wah Wu, Nov 02 2023

Formula

a(n) = n + A346870(n).

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