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A295423 a(n) = size of the part containing the diagonal of the symmetric representation of sigma(k) when k is the n-th number with an odd number of parts.

%I A295423 #13 Apr 12 2018 04:56:39
%S A295423 1,3,7,12,15,3,28,8,31,39,42,60,5,56,72,63,12,91,90,96,32,124,7,15,
%T A295423 120,120,168,16,127,144,36,195,18,186,9,224,180,234,20,252,21,20,217,
%U A295423 210,280,48,248,22,360,11,312,255,54,336,104,336,24,403,372,26,54,392,378,363,26,480,13,66
%N A295423 a(n) = size of the part containing the diagonal of the symmetric representation of sigma(k) when k is the n-th number with an odd number of parts.
%C A295423 a(n) = middle element of row A071562(n) in A237270.
%C A295423 Every odd prime number p occurs in the sequence since the symmetric representation of p^2 has the three parts ( (p^2 + 1)/2 , p, (p^2 + 1)/2 ).
%C A295423 Let n = 2^m * q, m>=0 and q odd, and let r = floor( (sqrt(8n+1) - 1)/2 ). The symmetric representation of sigma(n) has an odd number of parts precisely when there exists an odd divisor d of q such that d <= r and d*2^(m+1) > r (see the link in A071562).
%C A295423 Conjecture: Every even number, except 2, 4, 6, 10, and 14, occurs as the size of the middle part of the symmetric representation with an odd number of parts.
%C A295423 By symmetry, an even middle part must have an even width at the diagonal constraining the geometry for small areas bounded by adjacent Dyck paths.
%C A295423 Size 2 clearly is impossible with a pair of Dyck paths as boundary.
%C A295423 Size 4 is impossible since the two bounding Dyck paths must end in lengths of 2. However, the rightmost entry in a row of A237591 equals 2 only in rows k*(k+1)/2 - 1, for k>1, so no 2's exist at the diagonal in adjacent Dyck paths.
%C A295423 Size 6 is impossible if Conjecture (2) in A285356 is true since the middle area requires legs of length 3 in the Dyck path at the diagonal; and so are sizes 10 and 14 unless the middle part touches the previous part at a point.
%C A295423 Sizes 6, 10, and 14 do not occur as the middle part for any number n <= 35000 that has an odd number of parts in its symmetric representation of sigma.
%C A295423 For more information see A071562 and A237270.
%e A295423 a(4)  = 12 since A071562(4)  =  6 and row  6 of A237270 is (12).
%e A295423 a(8)  =  8 since A071562(8)  = 15 and row 15 of A237270 is (8, 8, 8).
%e A295423 a(23) =  7 since A071562(23) = 49 and row 49 of A237270 is (25, 7, 25).
%e A295423 a(16) = 63 since A071562(16) = 32 and row 32 of A237270 is (63).
%e A295423 Possible layouts for the middle of sizes 4, 6, 10 and 14 with possible Dyck path continuations:
%e A295423   area:  4          6             10               14
%e A295423                                               _|_ _
%e A295423                  _|_          _|_ _            |   |_ _
%e A295423      _|_ _        | |_         |   |_          |_      |
%e A295423       |   |       |   |_       |_    |_          |_    |_
%e A295423       |_ _|_      |_ _ _|_       |_    |           |_    |
%e A295423           |             |          |_ _|_            |_ _|_
%e A295423                                        |                 |
%t A295423 (* function a237270[] is defined in A237270 and the property *)
%t A295423 (* on odd divisors is used to minimize computations with a237270 *)
%t A295423 row[n_] := Floor[(Sqrt[8n+1]-1)/2]
%t A295423 evenExp[n_] := First[NestWhile[{#[[1]]+1, #[[2]]/2}&, {0,n}, EvenQ[Last[#]]&]]
%t A295423 oddSRQ[n_] := Module[{e=2^evenExp[n], r=row[n]}, Select[Divisors[n/e], #<=r&&2 e #>r&]!={}]
%t A295423 middle[list_] := list[[Ceiling[Length[list]/2]]]
%t A295423 a295423[m_, n_] := Map[middle[a237270[#]]&, Select[Range[m, n], oddSRQ]]
%t A295423 a295423[1, 170] (* data *)
%Y A295423 Subsequence of A295422.
%Y A295423 Cf. A071562, A237270, A237271, A237591, A237593, A285356.
%K A295423 nonn
%O A295423 1,2
%A A295423 _Hartmut F. W. Hoft_, Feb 12 2018