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

Author

Omar E. Pol, Oct 22 2017

Keywords

• nonn

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.

Examples

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

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.

Programs

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 *)

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

Formula

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

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

From Omar E. Pol, Nov 05 2017: (Start)

a(n) = A000203(n) + A024816(n) + A153485(n) - A004125(n).

a(n) = A000217(n) + A153485(n) - A004125(n).

a(n) = A000203(n) + A153485(n) + A244048(n). (End)

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