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%I #57 Sep 08 2022 08:46:06
%S 0,1,5,9,19,24,41,49,68,82,109,116,155,172,201,225,271,285,341,358,
%T 409,448,505,516,594,634,689,728,811,828,929,961,1041,1102,1177,1205,
%U 1331,1384,1465,1510,1639,1668,1805,1852,1947,2044,2161,2180,2344,2407
%N a(n) = n^2 - sigma(n).
%C From _Omar E. Pol_, Apr 11 2021: (Start)
%C If n is prime (A000040) then a(n) = n^2 - n - 1.
%C If n is a power of 2 (A000079) then a(n) = (n-1)^2.
%C If n is a perfect number (A000396) then a(n) = (n-1)^2 - 1, assuming there are no odd perfect numbers.
%C In order to construct the diagram of the symmetric representation of a(n) we use the following rules:
%C 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).
%C 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).
%C 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)
%H Harvey P. Dale, <a href="/A235799/b235799.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A000290(n) - A000203(n).
%F a(n) = A024916(n-1) + A004125(n), n > 1.
%F G.f.: x*(1 + x)/(1 - x)^3 - Sum_{k>=1} x^k/(1 - x^k)^2. - _Ilya Gutkovskiy_, Mar 17 2017
%F From _Omar E. Pol_, Apr 10 2021: (Start)
%F a(n) = A024816(n) + A000217(n-1).
%F a(n) = A067436(n) + A153485(n) + A244048(n). (End)
%e From _Omar E. Pol, Apr 04 2021: (Start)
%e Illustration of initial terms in the first quadrant for n = 1..6:
%e .
%e . y| _ _
%e . y| _ _ |_ _ _ |_ |
%e . y| _ |_ _ _| | | | |_|
%e . y| _ |_ _ |_| | _| | |_ _
%e . y| |_ _|_| | |_ | | | |
%e . y| |_ | | | | | | | |
%e . |_ _ |_|_ _ |_ _|_ _ |_ _ _|_ _ |_ _ _ _|_ _ |_ _ _ _ _|_ _
%e . x x x x x x
%e .
%e n: 1 2 3 4 5 6
%e a(n): 0 1 5 9 19 24
%e .
%e Illustration of initial terms in the first quadrant for n = 7..9:
%e . y| _ _ _ _
%e . y| _ _ _ |_ _ _ _ _| |
%e . y| _ _ _ |_ _ _ _ | | | _ _ |
%e . |_ _ _ _| | | | |_ | | |_ | |
%e . | | | |_ |_ _| | |_| _|
%e . | _| | |_ _ | |
%e . | | | | | |
%e . | | | | | |
%e . | | | | | |
%e . |_ _ _ _ _ _|_ _ |_ _ _ _ _ _ _|_ _ |_ _ _ _ _ _ _ _|_ _
%e . x x x
%e .
%e n: 7 8 9
%e a(n): 41 49 68
%e .
%e For n = 9 the figures 1, 2 and 3 below show respectively the three stages described in the Comments section as follows:
%e .
%e . y|_ _ _ _ _ 5 y|_ _ _ _ _ _ _ _ _ y| _ _ _ _
%e . |_ _ _ _ _| |_ _ _ _ _| | |_ _ _ _ _| |
%e . | |_ _ 3 | |_ _ R | | _ _ |
%e . | |_ | | |_ | | | |_ | |
%e . | |_|_ _ 5 | |_|_ _| | |_| _|
%e . | | | | | | | |
%e . | Q | | | Q | | | |
%e . | | | | | | | |
%e . | | | | | | | |
%e . |_ _ _ _ _ _ _ _|_|_ |_ _ _ _ _ _ _ _|_|_ |_ _ _ _ _ _ _ _|_ _
%e . x x x
%e . Figure 1. Figure 2. Figure 3.
%e . Symmetric Symmetric Symmetric
%e . representation representation representation
%e . of sigma(9) of sigma(9) of a(9) = 68
%e . A000203(9) = 13 A000203(9) = 13
%e . and of and of
%e . Q = A024916(8) = 56 R = A004125(9) = 12
%e . Q = A024916(8) = 56
%e .
%e 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)
%t Table[n^2-DivisorSigma[1,n],{n,50}] (* _Harvey P. Dale_, Sep 02 2016 *)
%o (PARI) vector(50, n, n^2 - sigma(n)) \\ _G. C. Greubel_, Oct 31 2018
%o (Magma) [n^2 - DivisorSigma(1,n): n in [1..50]]; // _G. C. Greubel_, Oct 31 2018
%Y Cf. A000040, A000079, A000203, A000217, A000290, A000396, A004125, A024816, A024916, A067436, A120444, A153485, A196020, A236104, A236112, A237593, A244048, A342344.
%K nonn
%O 1,3
%A _Omar E. Pol_, Jan 24 2014
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