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A235799
a(n) = n^2 - sigma(n).
4
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
1,3
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)
LINKS
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)
EXAMPLE
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)
MATHEMATICA
Table[n^2-DivisorSigma[1, n], {n, 50}] (* Harvey P. Dale, Sep 02 2016 *)
PROG
(PARI) vector(50, n, n^2 - sigma(n)) \\ G. C. Greubel, Oct 31 2018
(Magma) [n^2 - DivisorSigma(1, n): n in [1..50]]; // G. C. Greubel, Oct 31 2018
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 24 2014
STATUS
approved