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

1,2

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

Formula

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

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

Mathematica

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

Prog

(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

Crossrefs

Partial sums of A346878.

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

Sequence in context: A366102 A183905 A301288 * A009860 A294249 A138105

Adjacent sequences: A347151 A347152 A347153 * A347155 A347156 A347157

Keyword

nonn,easy

Author

Omar E. Pol, Aug 20 2021

Status

approved