A239053 Sum of divisors of 4*n-1.

4, 8, 12, 24, 20, 24, 40, 32, 48, 56, 44, 48, 72, 72, 60, 104, 68, 72, 124, 80, 84, 120, 112, 120, 156, 104, 108, 152, 144, 144, 168, 128, 132, 240, 140, 168, 228, 152, 192, 216, 164, 168, 260, 248, 180, 248, 216, 192, 336, 200, 240, 312, 212, 264, 296

Offset

1,1

Comments

Bisection of A008438.

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.

We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000203(4n-1) = A000203(A004767(n-1)).

a(n) = 4*A097723(n-1). - Joerg Arndt, Mar 09 2014

Sum_{k=1..n} a(k) = (Pi^2/4) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

Example

Illustration of initial terms:
-----------------------------------------------------
.        Branches of the spiral
.        in the third quadrant             n    a(n)
-----------------------------------------------------
.     _       _       _       _
.    | |     | |     | |     | |
.    | |     | |     | |     |_|_ _
.    | |     | |     | |    2  |_ _|       1      4
.    | |     | |     |_|_     2
.    | |     | |    4    |_
.    | |     |_|_ _        |_ _ _ _
.    | |    6      |_      |_ _ _ _|       2      8
.    |_|_ _ _        |_   4
.   8      | |_ _      |
.          |_    |     |_ _ _ _ _ _
.            |_  |_    |_ _ _ _ _ _|       3     12
.           8  |_ _|  6
.                  |
.                  |_ _ _ _ _ _ _ _
.                  |_ _ _ _ _ _ _ _|       4     24
.                 8
.
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.

Maple

A239053:=n->numtheory[sigma](4*n-1): seq(A239053(n), n=1..80); # Wesley Ivan Hurt, Dec 06 2016

Mathematica

DivisorSigma[1, 4*Range[60]-1] (* Harvey P. Dale, Dec 06 2016 *)
Table[DivisorSigma[1, 4 n - 1], {n, 100}] (* Vincenzo Librandi, Dec 07 2016 *)

Prog

(Magma) [SumOfDivisors(4*n-1): n in [1..60]]; // Vincenzo Librandi, Dec 07 2016
(PARI) a(n) = sigma(4*n-1); \\ Michel Marcus, Dec 07 2016

Crossrefs

Cf. A000203, A004767, A008438, A062731, A074400, A112610, A193553, A196020, A235791, A236104, A237270, A237591, A237593, A239050, A239052, A244050, A245092, A262626.

Sequence in context: A015781 A130643 A014617 * A272708 A157416 A278602

Adjacent sequences: A239050 A239051 A239052 * A239054 A239055 A239056

Keyword

nonn,easy

Author

Omar E. Pol, Mar 09 2014

Status

approved