%I #18 Apr 12 2018 04:56:21

%S 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,

%T 15,72,16,63,7,27,12,91,19,30,8,90,21,96,22,42,32,36,24,124,7,15,10,

%U 49,27,120,8,120,11,45,30,168,31,48,16,127,9,144,34,63,13,36,36,195,37,57,24

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

%C 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.

%C 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:

%C (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.

%C (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.

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

%e a(9) = 3; see the Example in A241558.

%e a(16) = 31 since its symmetric representation of sigma has one part of width one.

%e 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}.

%e a(50) = 15 since its symmetric representation of sigma has the three parts 39, 15, and 39.

%t (* function a237270 and its support are defined in A237270 *)

%t a295422[n_] := Module[{a=a237270[n]}, a[[Ceiling[Length[a]/2]]]]

%t Map[a295422, Range[75]] (* data *)

%Y Cf. A237270, A237271, A237593, A241558.

%K nonn

%O 1,2

%A _Hartmut F. W. Hoft_, Feb 12 2018