A346867 Sum of divisors of the numbers that have middle divisors.

1, 3, 7, 12, 15, 13, 28, 24, 31, 39, 42, 60, 31, 56, 72, 63, 48, 91, 90, 96, 78, 124, 57, 93, 120, 120, 168, 104, 127, 144, 144, 195, 96, 186, 121, 224, 180, 234, 112, 252, 171, 156, 217, 210, 280, 216, 248, 182, 360, 133, 312, 255, 252, 336, 240, 336, 168, 403, 372, 234

Offset

1,2

Comments

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is >= 1.

Also the width on the main diagonal equals the number of middle divisors.

So knowing this characteristic shape we can know if a number has middle divisors (or not) and the number of them just by looking at the diagram, even ignoring the concept of middle divisors.

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

For the definition of "width" see A249351.

Links

Table of n, a(n) for n=1..60.

Formula

a(n) = A000203(A071562(n)).

Example

a(4) = 12 because the sum of divisors of the fourth number that has middle divisors (i.e., 6) is 1 + 2 + 3 + 6 = 12.
On the other hand we can see that in the main diagonal of every diagram the width is >= 1 as shown below.
Illustration of initial terms:
m(n) = A071562(n).
.
   n   m(n) a(n)   Diagram
.                  _ _   _   _   _ _     _     _ _   _   _       _
   1    1    1    |_| | | | | | | | |   | |   | | | | | | |     | |
   2    2    3    |_ _|_| | | | | | |   | |   | | | | | | |     | |
                   _ _|  _|_| | | | |   | |   | | | | | | |     | |
   3    4    7    |_ _ _|    _|_| | |   | |   | | | | | | |     | |
                   _ _ _|  _|  _ _|_|   | |   | | | | | | |     | |
   4    6   12    |_ _ _ _|  _| |  _ _ _| |   | | | | | | |     | |
                   _ _ _ _| |_ _|_|    _ _|   | | | | | | |     | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | |     | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | |     | |
                              |  _ _|    _| |    _ _ _|_| |     | |
                   _ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|     | |
   7   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| |    _ _ _ _ _| |
                                  |  _ _|  _|    _|   |    _ _ _ _|
                   _ _ _ _ _ _ _ _| |     |     |  _ _|   |
   8   15   24    |_ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |
   9   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|
                   _ _ _ _ _ _ _ _ _| | |     |      _|
  10   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|
                   _ _ _ _ _ _ _ _ _ _| | |       |
  11   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|
                                          | |
                                          | |
                   _ _ _ _ _ _ _ _ _ _ _ _| |
  12   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _|
.
The n-th diagram has the property that at least it shares a vertex with the (n+1)-st diagram.

Mathematica

s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], Plus @@ d, 0]]; Select[Array[s, 150], # > 0 &] (* Amiram Eldar, Aug 19 2021 *)

Prog

(PARI) is(n) = fordiv(n, d, if(d^2>=n/2 && d^2<2*n, return(1))); 0 ; \\ A071562
apply(sigma, select(is, [1..200])) \\ Michel Marcus, Aug 19 2021

Crossrefs

Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, A240542, A245092, A249351, A262626, A281007, A299777, A346864.

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), A346868 (of numbers with no middle divisors).

Sequence in context: A310218 A295423 A339583 * A310219 A310220 A310221

Adjacent sequences: A346864 A346865 A346866 * A346868 A346869 A346870

Keyword

nonn

Author

Omar E. Pol, Aug 18 2021

Status

approved