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 A346868 #32 Aug 20 2021 00:22:52
%S A346868 4,6,8,18,12,14,24,18,20,32,36,24,42,40,30,32,48,54,38,60,56,42,44,84,
%T A346868 72,48,72,98,54,72,80,90,60,62,96,84,68,126,96,72,74,114,124,140,168,
%U A346868 80,126,84,108,132,120,90,168,128,144,120,98,102,216,104,192,162,108,110
%N A346868 Sum of divisors of the numbers with no middle divisors.
%C A346868 The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is equal to zero.
%C A346868 So knowing this characteristic shape we can know if a number has middle divisors (or not) just by looking at the diagram, even ignoring the concept of middle divisors.
%C A346868 Therefore we can see a geometric pattern of the distribution of the numbers with no middle divisors in the stepped pyramid described in A245092.
%C A346868 For the definition of "width" see A249351.
%C A346868 All terms are even numbers.
%F A346868 a(n) = A000203(A071561(n)).
%e A346868 a(4) = 18 because the sum of divisors of the fourth number with no middle divisors (i.e., 10) is 1 + 2 + 5 + 10 = 18.
%e A346868 On the other hand we can see that in the main diagonal of every diagram the width is equal to zero as shown below.
%e A346868 Illustration of initial terms:
%e A346868 m(n) = A071561(n).
%e A346868 .
%e A346868 n m(n) a(n) Diagram
%e A346868 . _ _ _ _ _ _ _ _ _ _ _ _ _
%e A346868 | | | | | | | | | | | | | | | | | | | | | |
%e A346868 _ _|_| | | | | | | | | | | | | | | | | | | | |
%e A346868 1 3 4 |_ _| _|_| | | | | | | | | | | | | | | | | | |
%e A346868 _ _ _| _|_| | | | | | | | | | | | | | | | |
%e A346868 2 5 6 |_ _ _| _| _ _| | | | | | | | | | | | | | | |
%e A346868 _ _ _ _| | _ _|_| | | | | | | | | | | | | |
%e A346868 3 7 8 |_ _ _ _| _ _|_| _ _|_| | | | | | | | | | | |
%e A346868 | _| | _ _ _| | | | | | | | | | |
%e A346868 _ _ _ _ _| | _|_| _ _ _|_| | | | | | | | |
%e A346868 4 10 18 |_ _ _ _ _ _| _ _| | _ _ _|_| | | | | | |
%e A346868 5 11 12 |_ _ _ _ _ _| | _| _| | _ _ _ _|_| | | | |
%e A346868 _ _ _ _ _ _ _| | _| _ _| | | _ _ _ _|_| | |
%e A346868 6 13 14 |_ _ _ _ _ _ _| | _ _| _| _| | | _ _ _ _ _| |
%e A346868 7 14 24 |_ _ _ _ _ _ _ _| | | | _|_| | _ _ _ _ _|
%e A346868 | _ _| _ _|_| | |
%e A346868 _ _ _ _ _ _ _ _ _| | _ _| _| _|_|
%e A346868 8 17 18 |_ _ _ _ _ _ _ _ _| | |_ _ _| |
%e A346868 _ _ _ _ _ _ _ _ _ _| | _ _| _|
%e A346868 9 19 20 |_ _ _ _ _ _ _ _ _ _| | | _ _|
%e A346868 _ _ _ _ _ _ _ _ _ _ _| | _ _ _|
%e A346868 10 21 32 |_ _ _ _ _ _ _ _ _ _ _| | | _ _|
%e A346868 11 22 36 |_ _ _ _ _ _ _ _ _ _ _ _| | |
%e A346868 12 23 24 |_ _ _ _ _ _ _ _ _ _ _ _| | |
%e A346868 | |
%e A346868 _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e A346868 13 26 42 |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e A346868 .
%t A346868 s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], 0, Plus @@ d]]; Select[Array[s, 110], # > 0 &] (* _Amiram Eldar_, Aug 19 2021 *)
%o A346868 (PARI) is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1; \\ A071561 apply(sigma, select(is, [1..150])) \\ _Michel Marcus_, Aug 19 2021
%Y A346868 Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, A245092, A249351, A262626, A281007, A299777, A346864.
%Y A346868 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).
%K A346868 nonn
%O A346868 1,1
%A A346868 _Omar E. Pol_, Aug 18 2021