A327329 Twice the sum of all divisors of all positive integers <= n.

2, 8, 16, 30, 42, 66, 82, 112, 138, 174, 198, 254, 282, 330, 378, 440, 476, 554, 594, 678, 742, 814, 862, 982, 1044, 1128, 1208, 1320, 1380, 1524, 1588, 1714, 1810, 1918, 2014, 2196, 2272, 2392, 2504, 2684, 2768, 2960, 3048, 3216, 3372, 3516, 3612, 3860, 3974, 4160, 4304, 4500, 4608, 4848, 4992

Offset: 1

Author

Omar E. Pol, Sep 25 2019

Keywords

• nonn

Comments

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).

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

Examples

			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.
                           _ _ _ _ _
                          |         |
                          |         |_
                          |           |_ _
                          |               |
                          |     56        |
                          |               |
                          |               |
           _ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _|
          |               |
          |               |
          |               |
          |       56      |
          |_ _            |
              |_          |
                |         |
                |_ _ _ _ _|
		

Links

• Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Crossrefs

Partial sums of A074400.

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

Programs

Mathematica

Accumulate[2*DivisorSigma[1,Range[60]]] (* Harvey P. Dale, Sep 25 2021 *)

PARI

a(n) = 2*sum(k=1, n, sigma(k)); \\ Michel Marcus, Dec 20 2021

Python

from sympy import divisor_sigma
from itertools import accumulate
def f(, n): return  + 2*divisor_sigma(n, 1)
def aupton(terms): return list(accumulate(range(terms+1), f))[1:]
print(aupton(55)) # Michael S. Branicky, Dec 16 2021

Python

from math import isqrt
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

Formula

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

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

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

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

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