A244048 Antisigma(n) minus the sum of remainders of n mod k, for k = 1,2,3,...,n.

0, 0, 1, 2, 5, 6, 12, 13, 20, 24, 32, 33, 49, 50, 60, 69, 84, 85, 106, 107, 129, 140, 154, 155, 191, 197, 213, 226, 254, 255, 297, 298, 329, 344, 364, 377, 432, 433, 455, 472, 522, 523, 577, 578, 618, 651, 677, 678, 754, 762, 805, 826

Offset

1,4

Comments

For n > 1 a(n) is the sum of all aliquot parts of all positive integers < n. - Omar E. Pol, Mar 27 2021

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Formula

a(n) = A024816(n) - A004125(n).

a(n) = A000217(n) - A000203(n) - A004125(n).

a(n) = A024916(n) - A000203(n) - A000217(n-1).

a(n) = A000217(n) - A123327(n).

a(n) = A153485(n-1), n >= 2.

Example

From Omar E. Pol, Mar 27 2021: (Start)
The following diagrams show a square dissection into regions that are the symmetric representation of A000203, A004125, A153485 and this sequence.
In order to construct every diagram we use the following rules:
At stage 1 in the first quadrant of the square grid we draw the symmetric representation of sigma(n) using the two Dyck paths described in the rows n and n-1 of A237593.
At stage 2 we draw a pair of orthogonal line segments (if it's necessary) such that in the drawing appears totally formed a square n X n. The area of the region that is above the symmetric representation of sigma(n) equals A004125(n).
At stage 3 we draw a zig-zag path with line segments of length 1 from (0,n-1) to (n-1,0) such that appears a staircase with n-1 steps. The area of the region (or regions) that is below the symmetric representation of sigma(n) and above the staircase equals a(n).
At stage 4 we draw a copy of the symmetric representation of A004125(n) rotated 180 degrees such that one of its vertices is the point (0,0). The area of the region (or regions) that is above of this region and below the staircase equals A153485(n).
Illustration for n = 1..6:
.                                                                    _ _ _ _ _ _
.                                                     _ _ _ _ _     |_ _ _  |_ R|
.                                        _ _ _ _ R   |_ _S_|  R|    | |_T | S |_|
.                             _ _ _ R   |_ _  |_|    | |_  |_ _|    |   |_|_ _  |
.                    _ _     |_S_|_|    | |_|_S |    |_U_|_T | |    |_  U |_T | |
.             _ S   |_ S|   U|_|_|S|    |_ U|_| |    |   | |_|S|    | |_    |_| |
.            |_|    |_|_|    |_|_|_|    |_|_ _|_|    |_V_|_U_|_|    |_V_|_ _ _|_|
.                  U        V   U       V
.
n:            1       2         3           4             5               6
R: A004125    0       0         1           1             4               3
S: A000203    1       3         4           7             6              12
T: a(n)       0       0         1           2             5               6
U: A153485    0       1         2           5             6              12
V: A004125    0       0         1           1             4               3
.
Illustration for n = 7..9:
.                                                      _ _ _ _ _ _ _ _ _
.                                _ _ _ _ _ _ _ _      |_ _ _S_ _|       |
.            _ _ _ _ _ _ _      |_ _ _ _  |     |     | |_      |_ _ R  |
.           |_ _S_ _|     |     | |_    | |_ R  |     |   |_    |_ S|   |
.           | |_    |_ R  |     |   |_  |_S |_ _|     |     |_  T |_|_ _|
.           |   |_  T |_ _|     |     |_T |_ _  |     |_ _    |_      | |
.           |_ _  |_    | |     |_ _  U |_    | |     |   |  U  |_    | |
.           |   |_U |_  |S|     |   |_    |_  | |     |   |_ _    |_  |S|
.           |  V  |   |_| |     |  V  |     |_| |     |  V    |     |_| |
.           |_ _ _|_ _ _|_|     |_ _ _|_ _ _ _|_|     |_ _ _ _|_ _ _ _|_|
.
n:                 7                    8                      9
R: A004125         8                    8                     12
S: A000203         8                   15                     12
T: a(n)           12                   13                     20
U: A153485        13                   20                     24
V: A004125         8                    8                     12
.
Illustration for n = 10..12:
.                                                         _ _ _ _ _ _ _ _ _ _ _ _
.                              _ _ _ _ _ _ _ _ _ _ _     |_ _ _ _ _ _  |         |
.     _ _ _ _ _ _ _ _ _ _     |_ _ _S_ _ _|         |    | |_        | |_ _   R  |
.    |_ _ _S_ _  |       |    | |_        |      R  |    |   |_      |     |_    |
.    | |_      | |_  R   |    |   |_      |_        |    |     |_    |_  S   |   |
.    |   |_    |_ _|_    |    |     |_      |_      |    |       |_    |_    |_ _|
.    |     |_      | |_ _|    |       |_   T  |_ _ _|    |         |_ T  |_ _ _  |
.    |       |_ T  |_ _  |    |_ _ _    |_        | |    |_ _        |_        | |
.    |_ _      |_      | |    |     |_ U  |_      | |    |   |    U    |_      | |
.    |   |_ U    |_    |S|    |       |_    |_    |S|    |   |_          |_    | |
.    |     |_      |_  | |    |         |     |_  | |    |     |_ _        |_  | |
.    |  V    |       |_| |    |  V      |       |_| |    |  V      |         |_| |
.    |_ _ _ _|_ _ _ _ _|_|    |_ _ _ _ _|_ _ _ _ _|_|    |_ _ _ _ _|_ _ _ _ _ _|_|
.
n:            10                         11                          12
R: A004125    13                         22                          17
S: A000203    18                         12                          28
T: a(n)       24                         32                          33
U: A153485    32                         33                          49
V: A004125    13                         22                          17
.
Note that in the diagrams the symmetric representation of a(n) is the same as the symmetric representation of A153485(n-1) rotated 180 degrees.
The original examples (dated Jun 24 2014) were only the diagrams for n = 11 and n = 12. (End)

Mathematica

With[{r=Range[100]}, Join[{0}, Accumulate[DivisorSigma[1, r]-r]]] (* Paolo Xausa, Oct 16 2023 *)

Prog

(Python)
from math import isqrt
def A244048(n): return (-n*(n-1)-(s:=isqrt(n-1))**2*(s+1) + sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1, s+1)))>>1 # Chai Wah Wu, Oct 22 2023

Crossrefs

Also zero together with A153485.

Cf. A000203, A000290, A001065, A067439, A024816, A024916, A067439, A123327, A196020, A236104, A237270, A237271, A237593, A244049, A244251, A262626, A342344.

Sequence in context: A057518 A289206 A153485 * A309043 A023143 A085206

Adjacent sequences: A244045 A244046 A244047 * A244049 A244050 A244051

Keyword

nonn

Author

Omar E. Pol, Jun 23 2014

Status

approved