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.
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).
KEYWORD
nonn
AUTHOR
Omar E. Pol, Aug 18 2021
STATUS
approved