A224613 a(n) = sigma(6*n).

12, 28, 39, 60, 72, 91, 96, 124, 120, 168, 144, 195, 168, 224, 234, 252, 216, 280, 240, 360, 312, 336, 288, 403, 372, 392, 363, 480, 360, 546, 384, 508, 468, 504, 576, 600, 456, 560, 546, 744, 504, 728, 528, 720, 720, 672, 576, 819, 684, 868, 702, 840, 648

Offset: 1

Author

Zak Seidov, Apr 22 2013

Keywords

• nonn

Comments

Conjectures: sigma(6n) > sigma(6n - 1) and sigma(6n) > sigma(6n + 1).

Conjectures are false. Try prime 73961483429 for n. One finds sigma(6*73961483429) < sigma(6*73961483429+1). The number n = 105851369791 provides a counterexample for the other case. - T. D. Noe, Apr 22 2013

Sum of the divisors of the numbers k which have the property that the width associated to the vertex of the first (also the last) valley of the smallest Dyck path of the symmetric representation of sigma(k) is equal to 2 (see example). Other positive integers have width 0 or 1 associated to the mentioned valley. - Omar E. Pol, Aug 11 2021

Examples

			From _Omar E. Pol_, Aug 11 2021: (Start)
Illustration of initial terms:
----------------------------------------------------------------------
   n    6*n   a(n)    Diagram:  1           2           3           4
----------------------------------------------------------------------
                                _           _           _           _
                               | |         | |         | |         | |
                               | |         | |         | |         | |
                          * _ _| |         | |         | |         | |
                           |  _ _|         | |         | |         | |
                      _ _ _| |_|           | |         | |         | |
   1     6     12    |_ _ _ _|      * _ _ _| |         | |         | |
                                    _|  _ _ _|         | |         | |
                                * _|  _| |             | |         | |
                                 |  _|  _|    * _ _ _ _| |         | |
                                 | |_ _|       |  _ _ _ _|         | |
                      _ _ _ _ _ _| |          _| | |               | |
   2    12     28    |_ _ _ _ _ _ _|        _|  _|_|    * _ _ _ _ _| |
                                      * _ _|  _|         |  _ _ _ _ _|
                                       |  _ _|        _ _| | |
                                       | |_ _|      _|  _ _| |
                                       | |        _|  _|  _ _|
                      _ _ _ _ _ _ _ _ _| |       |  _|  _|
   3    18     39    |_ _ _ _ _ _ _ _ _ _|  * _ _| |  _|
                                             |  _ _| |
                                             | |_ _ _|
                                             | |
                                             | |
                      _ _ _ _ _ _ _ _ _ _ _ _| |
   4    24     60    |_ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the mentioned vertices are aligned on two straight lines that meet at point (3,3).
a(n) equals the area (also the number of cells) in the n-th diagram. (End)
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), A283118 (k=5), this sequence (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).

Cf. A000203 (sigma(n)), A053224 (n: sigma(n) < sigma(n+1)).

Cf. A067825 (even n: sigma(n)< sigma(n+1)).

Cf. A008588, A237593.

Programs

Mathematica

DivisorSigma[1,6*Range[60]] (* Harvey P. Dale, Apr 16 2016 *)

PARI

a(n)=sigma(6*n) \\ Charles R Greathouse IV, Apr 22 2013

Python

from sympy import divisor_sigma
def a(n):  return divisor_sigma(6*n)
print([a(n) for n in range(1, 54)]) # Michael S. Branicky, Dec 28 2021

Python

from math import prod
from collections import Counter
from sympy import factorint
def A224613(n): return prod((p**(e+1)-1)//(p-1) for p, e in (Counter(factorint(n))+Counter([2,3])).items()) # Chai Wah Wu, Sep 07 2023

Formula

a(n) = A000203(6n).

a(n) = A000203(A008588(n)). - Omar E. Pol, Aug 11 2021

Sum_{k=1..n} a(k) = (55*Pi^2/72) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

Extensions

Corrected by Harvey P. Dale, Apr 16 2016