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A235799
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a(n) = n^2 - sigma(n).
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4
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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
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OFFSET
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1,3
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COMMENTS
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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)
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LINKS
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FORMULA
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G.f.: x*(1 + x)/(1 - x)^3 - Sum_{k>=1} x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Mar 17 2017
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EXAMPLE
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From _Omar E. Pol, Apr 04 2021: (Start)
Illustration of initial terms in the first quadrant for n = 1..6:
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. y| _ _
. y| _ _ |_ _ _ |_ |
. y| _ |_ _ _| | | | |_|
. y| _ |_ _ |_| | _| | |_ _
. y| |_ _|_| | |_ | | | |
. y| |_ | | | | | | | |
. |_ _ |_|_ _ |_ _|_ _ |_ _ _|_ _ |_ _ _ _|_ _ |_ _ _ _ _|_ _
. x x x x x x
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n: 1 2 3 4 5 6
a(n): 0 1 5 9 19 24
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Illustration of initial terms in the first quadrant for n = 7..9:
. y| _ _ _ _
. y| _ _ _ |_ _ _ _ _| |
. y| _ _ _ |_ _ _ _ | | | _ _ |
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. | | | |_ |_ _| | |_| _|
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. |_ _ _ _ _ _|_ _ |_ _ _ _ _ _ _|_ _ |_ _ _ _ _ _ _ _|_ _
. x x x
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n: 7 8 9
a(n): 41 49 68
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For n = 9 the figures 1, 2 and 3 below show respectively the three stages described in the Comments section as follows:
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. y|_ _ _ _ _ 5 y|_ _ _ _ _ _ _ _ _ y| _ _ _ _
. |_ _ _ _ _| |_ _ _ _ _| | |_ _ _ _ _| |
. | |_ _ 3 | |_ _ R | | _ _ |
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. | |_|_ _ 5 | |_|_ _| | |_| _|
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. | Q | | | Q | | | |
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. |_ _ _ _ _ _ _ _|_|_ |_ _ _ _ _ _ _ _|_|_ |_ _ _ _ _ _ _ _|_ _
. 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
. and of and of
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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)
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MATHEMATICA
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Table[n^2-DivisorSigma[1, n], {n, 50}] (* Harvey P. Dale, Sep 02 2016 *)
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PROG
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(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
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CROSSREFS
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Cf. A000040, A000079, A000203, A000217, A000290, A000396, A004125, A024816, A024916, A067436, A120444, A153485, A196020, A236104, A236112, A237593, A244048, A342344.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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