A327329 - OEIS

%I #59 Oct 22 2023 16:36:55

%S 2,8,16,30,42,66,82,112,138,174,198,254,282,330,378,440,476,554,594,

%T 678,742,814,862,982,1044,1128,1208,1320,1380,1524,1588,1714,1810,

%U 1918,2014,2196,2272,2392,2504,2684,2768,2960,3048,3216,3372,3516,3612,3860,3974,4160,4304,4500,4608,4848,4992

%N Twice the sum of all divisors of all positive integers <= n.

%C 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 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 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (first 16 levels)</a>

%F a(n) = 2*A024916(n).

%F a(n) = A243980(n)/2.

%F a(n) = A006218(n) + A222548(n).

%F a(n) = A001105(n) - A067436(n).

%F lim_{n->infinity} a(n)/(n^2) = Pi^2/6 = zeta(2) (cf. A013661). - _Omar E. Pol_, Dec 16 2021

%e 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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%t Accumulate[2*DivisorSigma[1,Range[60]]] (* _Harvey P. Dale_, Sep 25 2021 *)

%o (Python)

%o from sympy import divisor_sigma

%o from itertools import accumulate

%o def f(_, n): return _ + 2*divisor_sigma(n, 1)

%o def aupton(terms): return list(accumulate(range(terms+1), f))[1:]

%o print(aupton(55)) # _Michael S. Branicky_, Dec 16 2021

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

%o (Python)

%o from math import isqrt

%o 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 Partial sums of A074400.

%Y Cf. A001105, A006218, A013661, A024916, A067436, A222548, A236104, A237591, A237593, A243980, A245092, A262626.

%K nonn

%O 1,1

%A _Omar E. Pol_, Sep 25 2019