1,1

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

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

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

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/4) * exp(3*Pi/4) * Pi * 2^(1/4) / Gamma(3/4)^4 = A388827. - Simon Plouffe, Sep 18 2025

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.

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

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 *)

(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

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

nonn,easy

Omar E. Pol, Mar 09 2014

approved