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A326124 a(n) is the sum of all divisors of the first n positive even numbers.

%I A326124 #43 Oct 22 2023 00:48:38
%S A326124 3,10,22,37,55,83,107,138,177,219,255,315,357,413,485,548,602,693,753,
%T A326124 843,939,1023,1095,1219,1312,1410,1530,1650,1740,1908,2004,2131,2275,
%U A326124 2401,2545,2740,2854,2994,3162,3348,3474,3698,3830,4010,4244,4412,4556,4808,4979,5196,5412,5622,5784,6064,6280
%N A326124 a(n) is the sum of all divisors of the first n positive even numbers.
%C A326124 A326123(n)/a(n) converges to 3/5.
%C A326124 a(n) is also the total area of the terraces of the first n even-indexed levels of the stepped pyramid described in A245092.
%H A326124 Robert Israel, <a href="/A326124/b326124.txt">Table of n, a(n) for n = 1..10000</a>
%F A326124 a(n) = A024916(2n) - A326123(n).
%F A326124 a(n) ~ 5 * Pi^2 * n^2 / 24. - _Vaclav Kotesovec_, Aug 18 2021
%e A326124 For n = 3 the first three positive even numbers are [2, 4, 6] and their divisors are [1, 2], [1, 2, 4], [1, 2, 3, 6] respectively, and the sum of these divisors is 1 + 2 + 1 + 2 + 4 + 1 + 2 + 3 + 6 = 22, so a(3) = 22.
%p A326124 ListTools:-PartialSums(map(numtheory:-sigma, [seq(i,i=2..200,2)])); # _Robert Israel_, Jun 12 2019
%t A326124 Accumulate@ DivisorSigma[1, Range[2, 110, 2]] (* _Michael De Vlieger_, Jun 09 2019 *)
%o A326124 (PARI) terms(n) = my(s=0, i=0); for(k=1, n-1, if(i>=n, break); s+=sigma(2*k); print1(s, ", "); i++)
%o A326124 /* Print initial 50 terms as follows: */
%o A326124 terms(50) \\ _Felix Fröhlich_, Jun 08 2019
%o A326124 (PARI) a(n) = sum(k=1, 2*n, if (!(k%2), sigma(k))); \\ _Michel Marcus_, Jun 08 2019
%o A326124 (Python)
%o A326124 from math import isqrt
%o A326124 def A326124(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) # _Chai Wah Wu_, Oct 21 2023
%Y A326124 Partial sums of A062731.
%Y A326124 Cf. A000203, A005843, A024916, A237593, A245092, A299174, A326123.
%K A326124 nonn,easy
%O A326124 1,1
%A A326124 _Omar E. Pol_, Jun 07 2019