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

%I A346870 #35 Nov 02 2023 16:19:56
%S A346870 0,2,7,13,20,35,44,58,78,99,112,147,162,189,230,260,279,333,354,403,
%T A346870 456,495,520,595,637,682,747,810,841,948,981,1043,1120,1177,1250,1372,
%U A346870 1411,1474,1563,1668,1711,1850,1895,1986,2129,2204,2253,2408,2480,2596,2709,2814
%N A346870 Sum of all divisors, except the smallest and the largest of every number, of the first n positive even numbers.
%C A346870 Partial sums of the even-indexed terms of Chowla's function A048050.
%C A346870 a(n) has a symmetric representation.
%H A346870 Amiram Eldar, <a href="/A346870/b346870.txt">Table of n, a(n) for n = 1..10000</a>
%F A346870 a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - _Amiram Eldar_, May 15 2023
%p A346870 a:= proc(n) option remember; `if`(n=0, 0,
%p A346870       a(n-1)+numtheory[sigma](2*n)-1-2*n)
%p A346870     end:
%p A346870 seq(a(n), n=1..60);  # _Alois P. Heinz_, Aug 19 2021
%t A346870 s[n_] := DivisorSigma[1, 2*n] - 2*n - 1; Accumulate @ Array[s, 50] (* _Amiram Eldar_, Aug 19 2021 *)
%o A346870 (Python)
%o A346870 from sympy import divisors
%o A346870 from itertools import accumulate
%o A346870 def A346880(n): return sum(divisors(2*n)[1:-1])
%o A346870 def aupton(nn): return list(accumulate(A346880(n) for n in range(1, nn+1)))
%o A346870 print(aupton(52)) # _Michael S. Branicky_, Aug 19 2021
%o A346870 (Python)
%o A346870 from math import isqrt
%o A346870 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
%Y A346870 Partial sums of A346880.
%Y A346870 Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346869.
%K A346870 nonn,easy
%O A346870 1,2
%A A346870 _Omar E. Pol_, Aug 18 2021