A294016 a(n) = sum of all divisors of all positive integers <= n, minus the sum of remainders of n mod k, for k = 1, 2, 3, ..., n.

1, 4, 7, 14, 17, 30, 33, 48, 57, 74, 77, 110, 113, 134, 153, 184, 187, 230, 233, 278, 301, 330, 333, 406, 419, 452, 479, 536, 539, 624, 627, 690, 721, 762, 789, 900, 903, 948, 983, 1084, 1087, 1196, 1199, 1280, 1347, 1400, 1403, 1556, 1573, 1660, 1703, 1796, 1799, 1932, 1967, 2096, 2143, 2208, 2211, 2428, 2431, 2500

Offset

1,2

Comments

a(n) is also the area (also the number of cells) of the n-th polygon formed by the Dyck path described in A237593 and its mirror, as shown below in the example.

a(n) is also the volume (and the number of cubes) in the n-th level (starting from the top) of the pyramid described in A294017.

Links

Table of n, a(n) for n=1..62.

Formula

a(n) = A024916(n) - A004125(n) = A000290(n) - A067436(n).

a(n) = A000203(n) + A024816(n) + A153485(n) - A004125(n) = A000217(n) + A153485(n) - A004125(n) = A000203(n) + A153485(n) + A244048(n). - Omar E. Pol, Nov 05 2017

a(n) = (Pi^2/6 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 30 2024

Example

Illustration of initial terms:
.
.   _ 1
.  |_|_ _ 4
.    |   |
.    |_ _|_ _   7
.        |   |_
.        |_    |
.          |_ _|_ _ _  14
.              |     |_
.              |       |
.              |_      |
.                |_ _ _|_ _ _
.                      |     |  17
.                      |     |_ _
.                      |_ _      |
.                          |     |
.                          |_ _ _|_ _ _ _
.                                |       |_  30
.                                |         |_
.                                |           |
.                                |_          |
.                                  |_        |
.                                    |_ _ _ _|_ _ _ _
.                                            |       |
.                                            |       |_  33
.                                            |         |_ _
.                                            |_ _          |
.                                                |_        |
.                                                  |       |
.                                                  |_ _ _ _|
.

Maple

A294016 := proc(n)
    A024916(n)-A004125(n) ;
end proc:
seq(A294016(n), n=1..80) ; # R. J. Mathar, Nov 07 2017

Mathematica

Accumulate[Table[2*(DivisorSigma[1, n] - n) + 1, {n, 1, 100}]] (* Amiram Eldar, Mar 30 2024 *)

Prog

(Python)
from math import isqrt
def A294016(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1, s+1))-n**2 # Chai Wah Wu, Oct 22 2023

Crossrefs

Partial sums of A294015.

Partial sums gives A294017.

Cf. A000203, A000217, A000290, A013661, A004125, A024816, A024916, A067436, A153485, A196020, A235791, A236104, A237591, A237593, A244048, A245092.

Sequence in context: A310911 A255649 A209978 * A344533 A310912 A310913

Adjacent sequences: A294013 A294014 A294015 * A294017 A294018 A294019

Keyword

nonn

Author

Omar E. Pol, Oct 22 2017

Status

approved