A347155 Sum of divisors of nontriangular numbers.

3, 7, 6, 8, 15, 13, 12, 28, 14, 24, 31, 18, 39, 20, 42, 36, 24, 60, 31, 42, 40, 30, 72, 32, 63, 48, 54, 48, 38, 60, 56, 90, 42, 96, 44, 84, 72, 48, 124, 57, 93, 72, 98, 54, 120, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 68, 126, 96, 144, 72, 195, 74, 114, 124, 140

Offset: 1

Author

Omar E. Pol, Aug 20 2021

Keywords

• nonn

Comments

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.

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.

Therefore we can see a geometric pattern of the distribution of the nontriangular numbers in the stepped pyramid described in A245092.

If both Dyck paths have peaks on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317303.

If both Dyck paths have valleys on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317304.

Examples

			a(6) = 13 because the sum of divisors of the 6th nontriangular (i.e., 9) is 1 + 3 + 9 = 13.
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.
Illustration of initial terms:
m(n) = A014132(n).
.
   n   m(n) a(n)   Diagram
.                    _   _ _   _ _ _   _ _ _ _   _ _ _ _ _   _ _ _ _ _ _
                   _| | | | | | | | | | | | | | | | | | | | | | | | | | |
   1    2    3    |_ _|_| | | | | | | | | | | | | | | | | | | | | | | | |
                   _ _|  _|_| | | | | | | | | | | | | | | | | | | | | | |
   2    4    7    |_ _ _|    _|_| | | | | | | | | | | | | | | | | | | | |
   3    5    6    |_ _ _|  _|  _ _|_| | | | | | | | | | | | | | | | | | |
                   _ _ _ _|  _| |  _ _|_| | | | | | | | | | | | | | | | |
   4    7    8    |_ _ _ _| |_ _|_|    _ _|_| | | | | | | | | | | | | | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | | | | | | | | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | | | | | | | | |
                   _ _ _ _ _ _|  _ _|    _| |    _ _ _|_| | | | | | | | |
   7   11   12    |_ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|_| | | | | | |
   8   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| | |  _ _ _ _|_| | | | |
   9   13   14    |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _|_| | |
  10   14   24    |_ _ _ _ _ _ _ _| |     |     |  _|_|   |  _ _ _ _ _|_|
                   _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       | | |
  11   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|_| |
  12   17   18    |_ _ _ _ _ _ _ _ _| | |_ _ _|      _| |  _ _|
  13   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|  _|_|
  14   19   20    |_ _ _ _ _ _ _ _ _ _| | |       |_ _|
  15   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|
                   _ _ _ _ _ _ _ _ _ _ _| | |  _ _| |
  16   22   36    |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _|
  17   23   24    |_ _ _ _ _ _ _ _ _ _ _ _| | |
  18   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  19   25   31    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  20   26   42    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
  21   27   40    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column m gives the nontriangular numbers.
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.
a(n) is also 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 3 + 3 = 6, so a(3) = 6.
For more information see A237593.
		

Crossrefs

Cf. A000203, A000217, A014132, A237591, A237593, A245092, A262626, A317303, A317304, A346873.

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).

Programs

Mathematica

Array[DivisorSigma[1,#+Round@Sqrt[2#]]&,100] (* Giorgos Kalogeropoulos, Aug 20 2021 *)

PARI

a(n) = sigma(n + round(sqrt(2*n))); \\ Michel Marcus, Aug 21 2021

Formula

a(n) = A000203(A014132(n)).