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
Keywords
Examples
From _Omar E. Pol_, Aug 11 2021: (Start)
Illustration of initial terms:
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n 6*n a(n) Diagram: 1 2 3 4
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_ _ _ _
| | | | | | | |
| | | | | | | |
* _ _| | | | | | | |
| _ _| | | | | | |
_ _ _| |_| | | | | | |
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
Programs
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Mathematica
DivisorSigma[1,6*Range[60]] (* Harvey P. Dale, Apr 16 2016 *)
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PARI
a(n)=sigma(6*n) \\ Charles R Greathouse IV, Apr 22 2013
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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
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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).
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
Comments