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 A224613 #62 Sep 08 2023 01:28:14
%S A224613 12,28,39,60,72,91,96,124,120,168,144,195,168,224,234,252,216,280,240,
%T A224613 360,312,336,288,403,372,392,363,480,360,546,384,508,468,504,576,600,
%U A224613 456,560,546,744,504,728,528,720,720,672,576,819,684,868,702,840,648
%N A224613 a(n) = sigma(6*n).
%C A224613 Conjectures: sigma(6n) > sigma(6n - 1) and sigma(6n) > sigma(6n + 1).
%C A224613 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
%C A224613 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
%H A224613 Seiichi Manyama, <a href="/A224613/b224613.txt">Table of n, a(n) for n = 1..10000</a>
%F A224613 a(n) = A000203(6n).
%F A224613 a(n) = A000203(A008588(n)). - _Omar E. Pol_, Aug 11 2021
%F A224613 Sum_{k=1..n} a(k) = (55*Pi^2/72) * n^2 + O(n*log(n)). - _Amiram Eldar_, Dec 16 2022
%e A224613 From _Omar E. Pol_, Aug 11 2021: (Start)
%e A224613 Illustration of initial terms:
%e A224613 ----------------------------------------------------------------------
%e A224613 n 6*n a(n) Diagram: 1 2 3 4
%e A224613 ----------------------------------------------------------------------
%e A224613 _ _ _ _
%e A224613 | | | | | | | |
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%e A224613 * _ _| | | | | | | |
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%e A224613 1 6 12 |_ _ _ _| * _ _ _| | | | | |
%e A224613 _| _ _ _| | | | |
%e A224613 * _| _| | | | | |
%e A224613 | _| _| * _ _ _ _| | | |
%e A224613 | |_ _| | _ _ _ _| | |
%e A224613 _ _ _ _ _ _| | _| | | | |
%e A224613 2 12 28 |_ _ _ _ _ _ _| _| _|_| * _ _ _ _ _| |
%e A224613 * _ _| _| | _ _ _ _ _|
%e A224613 | _ _| _ _| | |
%e A224613 | |_ _| _| _ _| |
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%e A224613 _ _ _ _ _ _ _ _ _| | | _| _|
%e A224613 3 18 39 |_ _ _ _ _ _ _ _ _ _| * _ _| | _|
%e A224613 | _ _| |
%e A224613 | |_ _ _|
%e A224613 | |
%e A224613 | |
%e A224613 _ _ _ _ _ _ _ _ _ _ _ _| |
%e A224613 4 24 60 |_ _ _ _ _ _ _ _ _ _ _ _ _|
%e A224613 .
%e A224613 Note that the mentioned vertices are aligned on two straight lines that meet at point (3,3).
%e A224613 a(n) equals the area (also the number of cells) in the n-th diagram. (End)
%t A224613 DivisorSigma[1,6*Range[60]] (* _Harvey P. Dale_, Apr 16 2016 *)
%o A224613 (PARI) a(n)=sigma(6*n) \\ _Charles R Greathouse IV_, Apr 22 2013
%o A224613 (Python)
%o A224613 from sympy import divisor_sigma
%o A224613 def a(n): return divisor_sigma(6*n)
%o A224613 print([a(n) for n in range(1, 54)]) # _Michael S. Branicky_, Dec 28 2021
%o A224613 (Python)
%o A224613 from math import prod
%o A224613 from collections import Counter
%o A224613 from sympy import factorint
%o A224613 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
%Y A224613 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).
%Y A224613 Cf. A000203 (sigma(n)), A053224 (n: sigma(n) < sigma(n+1)).
%Y A224613 Cf. A067825 (even n: sigma(n)< sigma(n+1)).
%Y A224613 Cf. A008588, A237593.
%K A224613 nonn
%O A224613 1,1
%A A224613 _Zak Seidov_, Apr 22 2013
%E A224613 Corrected by _Harvey P. Dale_, Apr 16 2016