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

%I A347154 #30 Nov 02 2023 15:34:25
%S A347154 1,4,10,17,25,41,51,66,87,109,123,159,175,203,245,276,296,351,373,423,
%T A347154 477,517,543,619,662,708,774,838,870,978,1012,1075,1153,1211,1285,
%U A347154 1408,1448,1512,1602,1708,1752,1892,1938,2030,2174,2250,2300,2456,2529,2646,2760
%N A347154 Sum of all divisors, except the largest of every number, of the first n positive even numbers.
%C A347154 Sum of all aliquot divisors (or aliquot parts) of the first n positive even numbers.
%C A347154 Partial sums of the even-indexed terms of A001065.
%C A347154 a(n) has a symmetric representation.
%H A347154 Amiram Eldar, <a href="/A347154/b347154.txt">Table of n, a(n) for n = 1..10000</a>
%F A347154 a(n) = n + A346870(n).
%F A347154 a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - _Amiram Eldar_, May 15 2023
%t A347154 s[n_] := DivisorSigma[1, 2*n] - 2*n; Accumulate @ Array[s, 100] (* _Amiram Eldar_, Aug 20 2021 *)
%o A347154 (PARI) a(n) = sum(k=1, n, k*=2; sigma(k)-k); \\ _Michel Marcus_, Aug 20 2021
%o A347154 (Python)
%o A347154 from sympy import divisors
%o A347154 from itertools import accumulate
%o A347154 def A346878(n): return sum(divisors(2*n)[:-1])
%o A347154 def aupton(nn): return list(accumulate(A346878(n) for n in range(1, nn+1)))
%o A347154 print(aupton(51)) # _Michael S. Branicky_, Aug 20 2021
%o A347154 (Python)
%o A347154 from math import isqrt
%o A347154 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 A347154 Partial sums of A346878.
%Y A347154 Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346870, A346877, A346880, A347153.
%K A347154 nonn,easy
%O A347154 1,2
%A A347154 _Omar E. Pol_, Aug 20 2021