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

1,1

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

Links

Table of n, a(n) for n=1..55.

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

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

Example

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      |
          |_ _            |
              |_          |
                |         |
                |_ _ _ _ _|

Mathematica

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

Prog

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

Crossrefs

Partial sums of A074400.

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

Sequence in context: A252594 A137882 A194643 * A136514 A077071 A187216

Adjacent sequences: A327326 A327327 A327328 * A327330 A327331 A327332

Keyword

nonn

Author

Omar E. Pol, Sep 25 2019

Status

approved