A239052 Sum of divisors of 4*n-2.

3, 12, 18, 24, 39, 36, 42, 72, 54, 60, 96, 72, 93, 120, 90, 96, 144, 144, 114, 168, 126, 132, 234, 144, 171, 216, 162, 216, 240, 180, 186, 312, 252, 204, 288, 216, 222, 372, 288, 240, 363, 252, 324, 360, 270, 336, 384, 360, 294, 468, 306, 312, 576

Offset

1,1

Comments

Bisection of A062731 (odd part).

a(n) is also the total number of cells in the n-th branch of the second quadrant of the spiral formed by the parts of the symmetric representation of sigma(4n-2). For the quadrants 1, 3, 4 see A112610, A239053, A193553. The spiral has been obtained according to the following way: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A237270, see example.

We can find the spiral on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

Links

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

Formula

a(n) = A000203(4n-2) = A000203(A016825(n-1)).

a(n) = 3*A008438(n-1). - Joerg Arndt, Mar 09 2014

Sum_{k=1..n} a(k) = (3*Pi^2/8) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

Example

Illustration of initial terms:
------------------------------------------------------
.        Branches of the spiral
.        in the second quadrant             n    a(n)
------------------------------------------------------
.
.                  _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|         4     24
.                 | |
.             12 _| |
.               |_ _|  _ _ _ _ _ _
.         12 _ _|     |  _ _ _ _ _|         3     18
.      _ _ _| |    9 _| |
.     |  _ _ _|  9 _|_ _|
.     | |      _ _| |      _ _ _ _
.     | |     |  _ _| 12 _|  _ _ _|         2     12
.     | |     | |      _|   |
.     | |     | |     |  _ _|
.     | |     | |     | |    3 _ _
.     | |     | |     | |     |  _|         1      3
.     |_|     |_|     |_|     |_|
.
For n = 4 the sum of divisors of 4*n-2 is 1 + 2 + 7 + 14 = A000203(14) = 24. On the other hand the parts of the symmetric representation of sigma(14) are [12, 12] and the sum of them is 12 + 12 = 24, equaling the sum of divisors of 14, so a(4) = 24.

Mathematica

a[n_] := DivisorSigma[1, 4*n - 2]; Array[a, 100] (* Amiram Eldar, Dec 17 2022 *)

Crossrefs

Cf. A000203, A008438, A016825, A062731, A074400, A112610, A193553, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239050, A239053, A244050, A245092, A262626.

Sequence in context: A234904 A284806 A125821 * A063229 A061564 A348169

Adjacent sequences: A239049 A239050 A239051 * A239053 A239054 A239055

Keyword

nonn,easy

Author

Omar E. Pol, Mar 09 2014

Status

approved