A347154 - OEIS

%I #30 Nov 02 2023 15:34:25

%S 1,4,10,17,25,41,51,66,87,109,123,159,175,203,245,276,296,351,373,423,

%T 477,517,543,619,662,708,774,838,870,978,1012,1075,1153,1211,1285,

%U 1408,1448,1512,1602,1708,1752,1892,1938,2030,2174,2250,2300,2456,2529,2646,2760

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

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

%C Partial sums of the even-indexed terms of A001065.

%C a(n) has a symmetric representation.

%H Amiram Eldar, <a href="/A347154/b347154.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - _Amiram Eldar_, May 15 2023

%t s[n_] := DivisorSigma[1, 2*n] - 2*n; Accumulate @ Array[s, 100] (* _Amiram Eldar_, Aug 20 2021 *)

%o (PARI) a(n) = sum(k=1, n, k*=2; sigma(k)-k); \\ _Michel Marcus_, Aug 20 2021

%o (Python)

%o from sympy import divisors

%o from itertools import accumulate

%o def A346878(n): return sum(divisors(2*n)[:-1])

%o def aupton(nn): return list(accumulate(A346878(n) for n in range(1, nn+1)))

%o print(aupton(51)) # _Michael S. Branicky_, Aug 20 2021

%o (Python)

%o from math import isqrt

%o 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

%Y Partial sums of A346878.

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

%K nonn,easy

%O 1,2

%A _Omar E. Pol_, Aug 20 2021