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%I #37 Dec 17 2022 04:23:18
%S 4,8,12,24,20,24,40,32,48,56,44,48,72,72,60,104,68,72,124,80,84,120,
%T 112,120,156,104,108,152,144,144,168,128,132,240,140,168,228,152,192,
%U 216,164,168,260,248,180,248,216,192,336,200,240,312,212,264,296
%N Sum of divisors of 4*n-1.
%C Bisection of A008438.
%C a(n) is also the total number of cells in the n-th branch of the third quadrant of the spiral formed by the parts of the symmetric representation of sigma(4n-1), see example. For the quadrants 1, 2, 4 see A112610, A239052, A193553. The spiral has been obtained according to the following way: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A237270.
%C We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - _Omar E. Pol_, Dec 06 2016
%H Amiram Eldar, <a href="/A239053/b239053.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A000203(4n-1) = A000203(A004767(n-1)).
%F a(n) = 4*A097723(n-1). - _Joerg Arndt_, Mar 09 2014
%F Sum_{k=1..n} a(k) = (Pi^2/4) * n^2 + O(n*log(n)). - _Amiram Eldar_, Dec 17 2022
%e Illustration of initial terms:
%e -----------------------------------------------------
%e . Branches of the spiral
%e . in the third quadrant n a(n)
%e -----------------------------------------------------
%e . _ _ _ _
%e . | | | | | | | |
%e . | | | | | | |_|_ _
%e . | | | | | | 2 |_ _| 1 4
%e . | | | | |_|_ 2
%e . | | | | 4 |_
%e . | | |_|_ _ |_ _ _ _
%e . | | 6 |_ |_ _ _ _| 2 8
%e . |_|_ _ _ |_ 4
%e . 8 | |_ _ |
%e . |_ | |_ _ _ _ _ _
%e . |_ |_ |_ _ _ _ _ _| 3 12
%e . 8 |_ _| 6
%e . |
%e . |_ _ _ _ _ _ _ _
%e . |_ _ _ _ _ _ _ _| 4 24
%e . 8
%e .
%e For n = 4 the sum of divisors of 4*n-1 is 1 + 3 + 5 + 15 = A000203(15) = 24. On the other hand the parts of the symmetric representation of sigma(15) are [8, 8, 8] and the sum of them is 8 + 8 + 8 = 24, equaling the sum of divisors of 15, so a(4) = 24.
%p A239053:=n->numtheory[sigma](4*n-1): seq(A239053(n), n=1..80); # _Wesley Ivan Hurt_, Dec 06 2016
%t DivisorSigma[1,4*Range[60]-1] (* _Harvey P. Dale_, Dec 06 2016 *)
%t Table[DivisorSigma[1, 4 n - 1], {n, 100}] (* _Vincenzo Librandi_, Dec 07 2016 *)
%o (Magma) [SumOfDivisors(4*n-1): n in [1..60]]; // _Vincenzo Librandi_, Dec 07 2016
%o (PARI) a(n) = sigma(4*n-1); \\ _Michel Marcus_, Dec 07 2016
%Y Cf. A000203, A004767, A008438, A062731, A074400, A112610, A193553, A196020, A235791, A236104, A237270, A237591, A237593, A239050, A239052, A244050, A245092, A262626.
%K nonn,easy
%O 1,1
%A _Omar E. Pol_, Mar 09 2014
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