A235799 a(n) = n^2 - sigma(n).

0, 1, 5, 9, 19, 24, 41, 49, 68, 82, 109, 116, 155, 172, 201, 225, 271, 285, 341, 358, 409, 448, 505, 516, 594, 634, 689, 728, 811, 828, 929, 961, 1041, 1102, 1177, 1205, 1331, 1384, 1465, 1510, 1639, 1668, 1805, 1852, 1947, 2044, 2161, 2180, 2344, 2407

Offset: 1

Author

Omar E. Pol, Jan 24 2014

Keywords

• nonn

Comments

From Omar E. Pol, Apr 11 2021: (Start)

If n is prime (A000040) then a(n) = n^2 - n - 1.

If n is a power of 2 (A000079) then a(n) = (n-1)^2.

If n is a perfect number (A000396) then a(n) = (n-1)^2 - 1, assuming there are no odd perfect numbers.

In order to construct the diagram of the symmetric representation of a(n) 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. The area of the region that is below the symmetric representation of sigma(n) equals A024916(n-1).

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 turn OFF the cells of the symmetric representation of sigma(n). Then we turn ON the rest of the cells that are in the square n X n. The result is that the ON cell form the diagram of the symmetric representation of a(n). See the Example section. (End)

Examples

			From _Omar E. Pol, Apr 04 2021: (Start)
Illustration of initial terms in the first quadrant for n = 1..6:
.
.                                                             y|        _ _
.                                              y|      _ _     |_ _ _  |_  |
.                                 y|      _     |_ _ _|   |    |     |   |_|
.                      y|    _     |_ _  |_|    |        _|    |     |_ _
.             y|        |_ _|_|    |   |_       |       |      |         |
.      y|      |_       |   |      |     |      |       |      |         |
.       |_ _   |_|_ _   |_ _|_ _   |_ _ _|_ _   |_ _ _ _|_ _   |_ _ _ _ _|_ _
.          x        x          x            x              x                x
.
n:        1       2         3           4             5               6
a(n):     0       1         5           9            19              24
.
Illustration of initial terms in the first quadrant for n = 7..9:
.                                                y|          _ _ _ _
.                          y|          _ _ _      |_ _ _ _ _|       |
.      y|        _ _ _      |_ _ _ _  |     |     |          _ _    |
.       |_ _ _ _|     |     |       | |_    |     |         |_  |   |
.       |             |     |       |_  |_ _|     |           |_|  _|
.       |            _|     |         |_ _        |               |
.       |           |       |             |       |               |
.       |           |       |             |       |               |
.       |           |       |             |       |               |
.       |_ _ _ _ _ _|_ _    |_ _ _ _ _ _ _|_ _    |_ _ _ _ _ _ _ _|_ _
.                      x                     x                       x
.
n:              7                    8                      9
a(n):          41                   49                     68
.
For n = 9 the figures 1, 2 and 3 below show respectively the three stages described in the Comments section as follows:
.
.   y|_ _ _ _ _ 5            y|_ _ _ _ _ _ _ _ _      y|          _ _ _ _
.    |_ _ _ _ _|              |_ _ _ _ _|       |      |_ _ _ _ _|       |
.    |         |_ _ 3         |         |_ _ R  |      |          _ _    |
.    |         |_  |          |         |_  |   |      |         |_  |   |
.    |           |_|_ _ 5     |           |_|_ _|      |           |_|  _|
.    |               | |      |               | |      |               |
.    |      Q        | |      |       Q       | |      |               |
.    |               | |      |               | |      |               |
.    |               | |      |               | |      |               |
.    |_ _ _ _ _ _ _ _|_|_     |_ _ _ _ _ _ _ _|_|_     |_ _ _ _ _ _ _ _|_ _
.                       x                        x                        x
.         Figure 1.                Figure 2.                Figure 3.
.         Symmetric                Symmetric                Symmetric
.       representation           representation           representation
.         of sigma(9)              of sigma(9)             of a(9) = 68
.       A000203(9) = 13          A000203(9) = 13
.           and of                   and of
.     Q = A024916(8) = 56      R = A004125(9) = 12
.                              Q = A024916(8) = 56
.
Note that the symmetric representation of a(9) contains a hole formed by three cells because these three cells were the central part of the symmetric representation of sigma(9). (End)
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A000079, A000203, A000217, A000290, A000396, A004125, A024816, A024916, A067436, A120444, A153485, A196020, A236104, A236112, A237593, A244048, A342344.

Programs

Magma

[n^2 - DivisorSigma(1,n): n in [1..50]]; // G. C. Greubel, Oct 31 2018

Mathematica

Table[n^2-DivisorSigma[1,n],{n,50}] (* Harvey P. Dale, Sep 02 2016 *)

PARI

vector(50, n, n^2 - sigma(n)) \\ G. C. Greubel, Oct 31 2018

Formula

a(n) = A000290(n) - A000203(n).

a(n) = A024916(n-1) + A004125(n), n > 1.

G.f.: x*(1 + x)/(1 - x)^3 - Sum_{k>=1} x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Mar 17 2017

From Omar E. Pol, Apr 10 2021: (Start)

a(n) = A024816(n) + A000217(n-1).

a(n) = A067436(n) + A153485(n) + A244048(n). (End)