This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A327329 #59 Oct 22 2023 16:36:55
%S A327329 2,8,16,30,42,66,82,112,138,174,198,254,282,330,378,440,476,554,594,
%T A327329 678,742,814,862,982,1044,1128,1208,1320,1380,1524,1588,1714,1810,
%U A327329 1918,2014,2196,2272,2392,2504,2684,2768,2960,3048,3216,3372,3516,3612,3860,3974,4160,4304,4500,4608,4848,4992
%N A327329 Twice the sum of all divisors of all positive integers <= n.
%C A327329 a(n) has a symmetric representation. Using two opposite quadrants, where in each quadrant there is the Dyck path related to partitions described in the n-th row of triangle A237593, a(n) is the total area (or the total number of cells) of the structure (see the example).
%C A327329 a(n) is also the total area of the horizontal faces in the stepped pyramid with n levels described in A245092 (that is the total area of the terraces plus the area of the base). - _Omar E. Pol_, Dec 15 2021
%H A327329 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (first 16 levels)</a>
%F A327329 a(n) = 2*A024916(n).
%F A327329 a(n) = A243980(n)/2.
%F A327329 a(n) = A006218(n) + A222548(n).
%F A327329 a(n) = A001105(n) - A067436(n).
%F A327329 lim_{n->infinity} a(n)/(n^2) = Pi^2/6 = zeta(2) (cf. A013661). - _Omar E. Pol_, Dec 16 2021
%e A327329 Illustration of a(8) = 112 using a symmetric structure constructed with the Dyck path related to partitions described in the 8th row of triangle A237593.
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%e A327329 | 56 |
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%t A327329 Accumulate[2*DivisorSigma[1,Range[60]]] (* _Harvey P. Dale_, Sep 25 2021 *)
%o A327329 (Python)
%o A327329 from sympy import divisor_sigma
%o A327329 from itertools import accumulate
%o A327329 def f(_, n): return _ + 2*divisor_sigma(n, 1)
%o A327329 def aupton(terms): return list(accumulate(range(terms+1), f))[1:]
%o A327329 print(aupton(55)) # _Michael S. Branicky_, Dec 16 2021
%o A327329 (PARI) a(n) = 2*sum(k=1, n, sigma(k)); \\ _Michel Marcus_, Dec 20 2021
%o A327329 (Python)
%o A327329 from math import isqrt
%o A327329 def A327329(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)) # _Chai Wah Wu_, Oct 22 2023
%Y A327329 Partial sums of A074400.
%Y A327329 Cf. A001105, A006218, A013661, A024916, A067436, A222548, A236104, A237591, A237593, A243980, A245092, A262626.
%K A327329 nonn
%O A327329 1,1
%A A327329 _Omar E. Pol_, Sep 25 2019