A346866 Sum of divisors of the n-th second hexagonal number.
4, 18, 32, 91, 72, 168, 192, 270, 260, 576, 288, 868, 560, 720, 768, 1488, 864, 1482, 1120, 1764, 1408, 2808, 1152, 3420, 2232, 2268, 2880, 4480, 1800, 4464, 3328, 5292, 3264, 5184, 3456, 6734, 4712, 5760, 4480, 10890, 3528, 10368, 5280, 7560, 8736, 9216, 5760, 12152
Offset: 1
Examples
a(3) = 32 because the sum of divisors of the third second hexagonal number (i.e., 21) is 1 + 3 + 7 + 21 = 32.
On the other hand we can see that in the main diagonal of every diagram the smallest Dyck path has a peak and the largest Dyck path has a valley as shown below.
Illustration of initial terms:
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n h(n) a(n) Diagram
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_ _ _ _
| | | | | | | |
_ _|_| | | | | | |
1 3 4 |_ _| | | | | | |
| | | | | |
_ _| | | | | |
| _ _| | | | |
_ _|_| | | | |
| _| | | | |
_ _ _ _ _| | | | | |
2 10 18 |_ _ _ _ _ _| | | | |
_ _ _ _|_| | |
| | | |
_| | | |
| _| | |
_ _|_| | |
_ _| _| | |
|_ _ _| | |
| _ _ _ _ _ _ _| |
| | _ _ _ _ _ _|
_ _ _ _ _ _ _ _ _ _ _| | |
3 21 32 |_ _ _ _ _ _ _ _ _ _ _| _ _| |
| |
_| _ _|
| |
_ _| _|
_ _| _|
| _|
_ _ _| _ _|
| |
| _ _ _ _|
| |
| |
| |
| |
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
4 36 91 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column h gives the n-th second hexagonal number (A014105).
The widths of the main diagonal of the diagrams are [0, 0, 0, 1] respectively.
a(n) is the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 11 + 5 + 5 + 11 = 32, so a(3) = 32.
For n = 4 there is only one region (or part) of size 91 in the fourth diagram so a(4) = 91.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Crossrefs
Bisection of A074285.
Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors), A347155 (of nontriangular numbers).
Programs
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Mathematica
a[n_] := DivisorSigma[1, n*(2*n + 1)]; Array[a, 50] (* Amiram Eldar, Aug 18 2021 *)
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PARI
a(n) = sigma(n*(2*n + 1)); \\ Michel Marcus, Aug 18 2021
Formula
Sum_{k=1..n} a(k) ~ 4*n^3/3. - Amiram Eldar, Dec 31 2024
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