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 A346867 #40 Aug 20 2021 00:22:41
%S A346867 1,3,7,12,15,13,28,24,31,39,42,60,31,56,72,63,48,91,90,96,78,124,57,
%T A346867 93,120,120,168,104,127,144,144,195,96,186,121,224,180,234,112,252,
%U A346867 171,156,217,210,280,216,248,182,360,133,312,255,252,336,240,336,168,403,372,234
%N A346867 Sum of divisors of the numbers that have middle divisors.
%C A346867 The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is >= 1.
%C A346867 Also the width on the main diagonal equals the number of middle divisors.
%C A346867 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.
%C A346867 Therefore we can see a geometric pattern of the distribution of the numbers with middle divisors in the stepped pyramid described in A245092.
%C A346867 For the definition of "width" see A249351.
%F A346867 a(n) = A000203(A071562(n)).
%e A346867 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.
%e A346867 On the other hand we can see that in the main diagonal of every diagram the width is >= 1 as shown below.
%e A346867 Illustration of initial terms:
%e A346867 m(n) = A071562(n).
%e A346867 .
%e A346867 n m(n) a(n) Diagram
%e A346867 . _ _ _ _ _ _ _ _ _ _ _ _
%e A346867 1 1 1 |_| | | | | | | | | | | | | | | | | | | |
%e A346867 2 2 3 |_ _|_| | | | | | | | | | | | | | | | | |
%e A346867 _ _| _|_| | | | | | | | | | | | | | | |
%e A346867 3 4 7 |_ _ _| _|_| | | | | | | | | | | | | |
%e A346867 _ _ _| _| _ _|_| | | | | | | | | | | |
%e A346867 4 6 12 |_ _ _ _| _| | _ _ _| | | | | | | | | | |
%e A346867 _ _ _ _| |_ _|_| _ _| | | | | | | | | |
%e A346867 5 8 15 |_ _ _ _ _| _| | _ _ _|_| | | | | | | |
%e A346867 6 9 13 |_ _ _ _ _| | _|_| | _ _ _|_| | | | | |
%e A346867 | _ _| _| | _ _ _|_| | | |
%e A346867 _ _ _ _ _ _| | _| _| _| | _ _ _ _| | |
%e A346867 7 12 28 |_ _ _ _ _ _ _| |_ _| _| _ _| | _ _ _ _ _| |
%e A346867 | _ _| _| _| | _ _ _ _|
%e A346867 _ _ _ _ _ _ _ _| | | | _ _| |
%e A346867 8 15 24 |_ _ _ _ _ _ _ _| | _ _| _ _|_| |
%e A346867 9 16 31 |_ _ _ _ _ _ _ _ _| | _ _| _| _ _|
%e A346867 _ _ _ _ _ _ _ _ _| | | | _|
%e A346867 10 18 39 |_ _ _ _ _ _ _ _ _ _| | _ _| _|
%e A346867 _ _ _ _ _ _ _ _ _ _| | | |
%e A346867 11 20 42 |_ _ _ _ _ _ _ _ _ _ _| | _ _ _|
%e A346867 | |
%e A346867 | |
%e A346867 _ _ _ _ _ _ _ _ _ _ _ _| |
%e A346867 12 24 60 |_ _ _ _ _ _ _ _ _ _ _ _ _|
%e A346867 .
%e A346867 The n-th diagram has the property that at least it shares a vertex with the (n+1)-st diagram.
%t A346867 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 *)
%o A346867 (PARI) is(n) = fordiv(n, d, if(d^2>=n/2 && d^2<2*n, return(1))); 0 ; \\ A071562
%o A346867 apply(sigma, select(is, [1..200])) \\ _Michel Marcus_, Aug 19 2021
%Y A346867 Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, A240542, A245092, A249351, A262626, A281007, A299777, A346864.
%Y A346867 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).
%K A346867 nonn
%O A346867 1,2
%A A346867 _Omar E. Pol_, Aug 18 2021