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A347154
Sum of all divisors, except the largest of every number, of the first n positive even numbers.
4
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
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
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 20 2021
STATUS
approved