This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A346866 #68 Jan 15 2025 08:38:29
%S A346866 4,18,32,91,72,168,192,270,260,576,288,868,560,720,768,1488,864,1482,
%T A346866 1120,1764,1408,2808,1152,3420,2232,2268,2880,4480,1800,4464,3328,
%U A346866 5292,3264,5184,3456,6734,4712,5760,4480,10890,3528,10368,5280,7560,8736,9216,5760,12152
%N A346866 Sum of divisors of the n-th second hexagonal number.
%C A346866 The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the smallest Dyck path has a peak and the largest Dyck path has a valley.
%C A346866 So knowing this characteristic shape we can know if a number is a second hexagonal number (or not) just by looking at the diagram, even ignoring the concept of second hexagonal number.
%C A346866 Therefore we can see a geometric pattern of the distribution of the second hexagonal numbers in the stepped pyramid described in A245092.
%H A346866 Amiram Eldar, <a href="/A346866/b346866.txt">Table of n, a(n) for n = 1..10000</a>
%F A346866 a(n) = A000203(A014105(n)).
%F A346866 Sum_{k=1..n} a(k) ~ 4*n^3/3. - _Amiram Eldar_, Dec 31 2024
%e A346866 a(3) = 32 because the sum of divisors of the third second hexagonal number (i.e., 21) is 1 + 3 + 7 + 21 = 32.
%e A346866 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.
%e A346866 Illustration of initial terms:
%e A346866 ---------------------------------------------------------------------------------------
%e A346866 n h(n) a(n) Diagram
%e A346866 ---------------------------------------------------------------------------------------
%e A346866 _ _ _ _
%e A346866 | | | | | | | |
%e A346866 _ _|_| | | | | | |
%e A346866 1 3 4 |_ _| | | | | | |
%e A346866 | | | | | |
%e A346866 _ _| | | | | |
%e A346866 | _ _| | | | |
%e A346866 _ _|_| | | | |
%e A346866 | _| | | | |
%e A346866 _ _ _ _ _| | | | | |
%e A346866 2 10 18 |_ _ _ _ _ _| | | | |
%e A346866 _ _ _ _|_| | |
%e A346866 | | | |
%e A346866 _| | | |
%e A346866 | _| | |
%e A346866 _ _|_| | |
%e A346866 _ _| _| | |
%e A346866 |_ _ _| | |
%e A346866 | _ _ _ _ _ _ _| |
%e A346866 | | _ _ _ _ _ _|
%e A346866 _ _ _ _ _ _ _ _ _ _ _| | |
%e A346866 3 21 32 |_ _ _ _ _ _ _ _ _ _ _| _ _| |
%e A346866 | |
%e A346866 _| _ _|
%e A346866 | |
%e A346866 _ _| _|
%e A346866 _ _| _|
%e A346866 | _|
%e A346866 _ _ _| _ _|
%e A346866 | |
%e A346866 | _ _ _ _|
%e A346866 | |
%e A346866 | |
%e A346866 | |
%e A346866 | |
%e A346866 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e A346866 4 36 91 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e A346866 .
%e A346866 Column h gives the n-th second hexagonal number (A014105).
%e A346866 The widths of the main diagonal of the diagrams are [0, 0, 0, 1] respectively.
%e A346866 a(n) is the area (and the number of cells) of the n-th diagram.
%e A346866 For n = 3 the sum of the regions (or parts) of the third diagram is 11 + 5 + 5 + 11 = 32, so a(3) = 32.
%e A346866 For n = 4 there is only one region (or part) of size 91 in the fourth diagram so a(4) = 91.
%t A346866 a[n_] := DivisorSigma[1, n*(2*n + 1)]; Array[a, 50] (* _Amiram Eldar_, Aug 18 2021 *)
%o A346866 (PARI) a(n) = sigma(n*(2*n + 1)); \\ _Michel Marcus_, Aug 18 2021
%Y A346866 Bisection of A074285.
%Y A346866 Cf. A000203, A000384, A014105, A237591, A237593, A245092, A262626, A346864.
%Y A346866 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).
%K A346866 nonn,easy
%O A346866 1,1
%A A346866 _Omar E. Pol_, Aug 17 2021