A243980 Four times the sum of all divisors of all positive integers <= n.

4, 16, 32, 60, 84, 132, 164, 224, 276, 348, 396, 508, 564, 660, 756, 880, 952, 1108, 1188, 1356, 1484, 1628, 1724, 1964, 2088, 2256, 2416, 2640, 2760, 3048, 3176, 3428, 3620, 3836, 4028, 4392, 4544, 4784, 5008, 5368, 5536, 5920, 6096, 6432, 6744, 7032, 7224, 7720

Offset

1,1

Comments

Also number of "ON" cells at n-th stage in a structure which looks like a simple 2-dimensional cellular automaton (see example). The structure is formed by the reflection on the four quadrants from the diagram of the symmetry of sigma in the first quadrant after n-th stage, hence the area in each quadrant equals the area of each wedge and equals A024916(n); the sum of all divisors of all positive integers <= n. For more information about the diagram see A237593 and A237270.

Links

Indranil Ghosh, Table of n, a(n) for n = 1..7342

Formula

a(n) = A016742(n) - 4*A004125(n) = 4*A024916(n).

a(n) = 2*(A006218(n) + A222548(n)) = 2*A327329(n). - Omar E. Pol, Sep 25 2019

Example

Illustration of the structure after 16 stages (contains 880 ON cells):
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.

Mathematica

Accumulate[4*DivisorSigma[1, Range[50]]] (* Harvey P. Dale, May 13 2018 *)

Prog

(Python)
from math import isqrt
def A243980(n): return -(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

Crossrefs

Partial sums of A239050.

Partial sums give A244050.

Cf. A000203, A000290, A004125, A016742, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934.

Sequence in context: A239611 A359195 A031050 * A119677 A326873 A126032

Adjacent sequences: A243977 A243978 A243979 * A243981 A243982 A243983

Keyword

nonn

Author

Omar E. Pol, Jun 18 2014

Status

approved