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 A347155 #28 Aug 21 2021 16:05:42 %S A347155 3,7,6,8,15,13,12,28,14,24,31,18,39,20,42,36,24,60,31,42,40,30,72,32, %T A347155 63,48,54,48,38,60,56,90,42,96,44,84,72,48,124,57,93,72,98,54,120,120, %U A347155 80,90,60,168,62,96,104,127,84,68,126,96,144,72,195,74,114,124,140 %N A347155 Sum of divisors of nontriangular numbers. %C A347155 The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys. %C A347155 So knowing this characteristic shape we can know if a number is a nontriangular number (or not) just by looking at the diagram, even ignoring the concept of nontriangular number. %C A347155 Therefore we can see a geometric pattern of the distribution of the nontriangular numbers in the stepped pyramid described in A245092. %C A347155 If both Dyck paths have peaks on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317303. %C A347155 If both Dyck paths have valleys on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317304. %F A347155 a(n) = A000203(A014132(n)). %e A347155 a(6) = 13 because the sum of divisors of the 6th nontriangular (i.e., 9) is 1 + 3 + 9 = 13. %e A347155 On the other we can see that in the main diagonal of the diagrams both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys as shown below. %e A347155 Illustration of initial terms: %e A347155 m(n) = A014132(n). %e A347155 . %e A347155 n m(n) a(n) Diagram %e A347155 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A347155 _| | | | | | | | | | | | | | | | | | | | | | | | | | | %e A347155 1 2 3 |_ _|_| | | | | | | | | | | | | | | | | | | | | | | | | %e A347155 _ _| _|_| | | | | | | | | | | | | | | | | | | | | | | %e A347155 2 4 7 |_ _ _| _|_| | | | | | | | | | | | | | | | | | | | | %e A347155 3 5 6 |_ _ _| _| _ _|_| | | | | | | | | | | | | | | | | | | %e A347155 _ _ _ _| _| | _ _|_| | | | | | | | | | | | | | | | | %e A347155 4 7 8 |_ _ _ _| |_ _|_| _ _|_| | | | | | | | | | | | | | | %e A347155 5 8 15 |_ _ _ _ _| _| | _ _ _|_| | | | | | | | | | | | | %e A347155 6 9 13 |_ _ _ _ _| | _|_| | _ _ _|_| | | | | | | | | | | %e A347155 _ _ _ _ _ _| _ _| _| | _ _ _|_| | | | | | | | | %e A347155 7 11 12 |_ _ _ _ _ _| | _| _| _| | _ _ _ _|_| | | | | | | %e A347155 8 12 28 |_ _ _ _ _ _ _| |_ _| _| _ _| | | _ _ _ _|_| | | | | %e A347155 9 13 14 |_ _ _ _ _ _ _| | _ _| _| _| | | _ _ _ _|_| | | %e A347155 10 14 24 |_ _ _ _ _ _ _ _| | | | _|_| | _ _ _ _ _|_| %e A347155 _ _ _ _ _ _ _ _| | _ _| _ _|_| | | | %e A347155 11 16 31 |_ _ _ _ _ _ _ _ _| | _ _| _| _ _|_| | %e A347155 12 17 18 |_ _ _ _ _ _ _ _ _| | |_ _ _| _| | _ _| %e A347155 13 18 39 |_ _ _ _ _ _ _ _ _ _| | _ _| _| _|_| %e A347155 14 19 20 |_ _ _ _ _ _ _ _ _ _| | | |_ _| %e A347155 15 20 42 |_ _ _ _ _ _ _ _ _ _ _| | _ _ _| _| %e A347155 _ _ _ _ _ _ _ _ _ _ _| | | _ _| | %e A347155 16 22 36 |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _| %e A347155 17 23 24 |_ _ _ _ _ _ _ _ _ _ _ _| | | %e A347155 18 24 60 |_ _ _ _ _ _ _ _ _ _ _ _ _| | %e A347155 19 25 31 |_ _ _ _ _ _ _ _ _ _ _ _ _| | %e A347155 20 26 42 |_ _ _ _ _ _ _ _ _ _ _ _ _ _| %e A347155 21 27 40 |_ _ _ _ _ _ _ _ _ _ _ _ _ _| %e A347155 . %e A347155 Column m gives the nontriangular numbers. %e A347155 Also the diagrams have on the main diagonal the following property: diagram [1] has peaks, diagrams [2, 3] have valleys, diagrams [4, 5, 6] have peaks, diagrams [7, 8, 9, 10] have valleys, and so on. %e A347155 a(n) is also the area (and the number of cells) of the n-th diagram. %e A347155 For n = 3 the sum of the regions (or parts) of the third diagram is 3 + 3 = 6, so a(3) = 6. %e A347155 For more information see A237593. %t A347155 Array[DivisorSigma[1,#+Round@Sqrt[2#]]&,100] (* _Giorgos Kalogeropoulos_, Aug 20 2021 *) %o A347155 (PARI) a(n) = sigma(n + round(sqrt(2*n))); \\ _Michel Marcus_, Aug 21 2021 %Y A347155 Cf. A000203, A000217, A014132, A237591, A237593, A245092, A262626, A317303, A317304, A346873. %Y A347155 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), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors). %K A347155 nonn %O A347155 1,1 %A A347155 _Omar E. Pol_, Aug 20 2021