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 A346865 #49 Dec 10 2021 11:18:53
%S A346865 1,12,24,56,78,144,112,360,234,360,384,672,434,960,720,992,864,1872,
%T A346865 760,2352,1344,1584,1872,2880,1767,3024,2160,4032,2400,4320,1984,6552,
%U A346865 4032,3672,4608,6552,2812,7440,5376,7200,5082,8064,4752,10080,7020,8064,6144
%N A346865 Sum of divisors of the n-th hexagonal number.
%C A346865 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 valley and the largest Dyck path has a peak.
%C A346865 So knowing this characteristic shape we can know if a number is an hexagonal number (or not) just by looking at the diagram, even ignoring the concept of hexagonal number.
%C A346865 Therefore we can see a geometric pattern of the distribution of the hexagonal numbers in the stepped pyramid described in A245092.
%F A346865 a(n) = A000203(A000384(n)).
%F A346865 Sum_{k=1..n} a(k) ~ 4*n^3/3. - _Vaclav Kotesovec_, Aug 18 2021
%e A346865 a(3) = 24 because the sum of divisors of the third hexagonal number (i.e., 15) is 1 + 3 + 5 + 15 = 24.
%e A346865 On the other hand we can see that in the main diagonal of every diagram the smallest Dyck path has a valley and the largest Dyck path has a peak as shown below.
%e A346865 Illustration of initial terms:
%e A346865 -------------------------------------------------------------------------
%e A346865 n H(n) a(n) Diagram
%e A346865 -------------------------------------------------------------------------
%e A346865 _ _ _ _
%e A346865 1 1 1 |_| | | | | | |
%e A346865 | | | | | |
%e A346865 _ _| | | | | |
%e A346865 | _| | | | |
%e A346865 _ _ _| _| | | | |
%e A346865 2 6 12 |_ _ _ _| | | | |
%e A346865 | | | |
%e A346865 _ _ _|_| | |
%e A346865 _ _| | | |
%e A346865 | _| | |
%e A346865 _| _| | |
%e A346865 |_ _| | |
%e A346865 | | |
%e A346865 _ _ _ _ _ _ _ _| _ _ _ _ _| |
%e A346865 3 15 24 |_ _ _ _ _ _ _ _| | _ _ _ _ _|
%e A346865 | |
%e A346865 _ _| |
%e A346865 _ _| _ _|
%e A346865 | _|
%e A346865 _| _|
%e A346865 | _|
%e A346865 _ _ _| |
%e A346865 | _ _ _|
%e A346865 | |
%e A346865 | |
%e A346865 | |
%e A346865 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e A346865 4 28 56 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e A346865 .
%e A346865 Column H gives the nonzero hexagonal numbers (A000384).
%e A346865 a(n) is also the area (and the number of cells) of the n-th diagram.
%e A346865 For n = 3 the sum of the regions (or parts) of the third diagram is 8 + 8 + 8 = 24, so a(3) = 24.
%e A346865 For more information see A237593.
%t A346865 a[n_] := DivisorSigma[1, n*(2*n - 1)]; Array[a, 50] (* _Amiram Eldar_, Aug 18 2021 *)
%o A346865 (PARI) a(n) = sigma(n*(2*n-1)); \\ _Michel Marcus_, Aug 18 2021
%o A346865 (Python)
%o A346865 from sympy import divisors
%o A346865 def a(n): return sum(divisors(n*(2*n - 1)))
%o A346865 print([a(n) for n in range(1, 48)]) # _Michael S. Branicky_, Aug 20 2021
%Y A346865 Bisection of A074285.
%Y A346865 Cf. A000203, A000384, A237591, A237593, A245092, A262626, A346864.
%Y A346865 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), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors).
%K A346865 nonn
%O A346865 1,2
%A A346865 _Omar E. Pol_, Aug 17 2021