A295422 a(n) = size of the middle part, i.e., the part closest to or crossing the diagonal, in the symmetric representation of sigma(n).

1, 3, 2, 7, 3, 12, 4, 15, 3, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 5, 18, 12, 60, 5, 21, 6, 56, 15, 72, 16, 63, 7, 27, 12, 91, 19, 30, 8, 90, 21, 96, 22, 42, 32, 36, 24, 124, 7, 15, 10, 49, 27, 120, 8, 120, 11, 45, 30, 168, 31, 48, 16, 127, 9, 144, 34, 63, 13, 36, 36, 195, 37, 57, 24

Offset: 1

Author

Hartmut F. W. Hoft, Feb 12 2018

Keywords

• nonn

Comments

When the symmetric representation of sigma(n) has an even number of parts, e.g., for every prime, the two middle parts have equal size so either one may be chosen.

Since a(45) = 32 while A241558(45) = 23 the two sequences are different, indeed both respective complements of the sequences, A241558 in a and a in A241558, are infinite as the symmetric representations of the following two subsequences of this sequence show:

(1) n = 5*3^k, k>1, has the 3 parts ( (5*3^k + 1)/2, 4*(3^k - 1), (5*3^k + 1)/2 ) with the middle the largest part.

(2) n = p^2, p > 2 prime, has the 3 parts ( (p^2 + 1)/2, p, (p^2 + 1)/2 ) with the middle the smallest part.

The parts of the symmetric representation of sigma are in A237270.

Examples

			a(9) = 3; see the Example in A241558.
a(16) = 31 since its symmetric representation of sigma has one part of width one.
a(41) = 21 since for any odd prime number p, row p of A237270 consists of the two parts: {(p+1)/2, (p+1)/2}.
a(50) = 15 since its symmetric representation of sigma has the three parts 39, 15, and 39.
		

Crossrefs

Cf. A237270, A237271, A237593, A241558.

Programs

Mathematica

(* function a237270 and its support are defined in A237270 *)
a295422[n_] := Module[{a=a237270[n]}, a[[Ceiling[Length[a]/2]]]]
Map[a295422, Range[75]] (* data *)