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A361905 Numbers k for which sqrt(k/2) divides k and the width at the diagonal of the symmetric representation of sigma(k) equals 1.

%I A361905 #11 May 05 2023 19:38:03
%S A361905 2,8,18,32,50,98,128,162,200,242,338,392,512,578,722,882,968,1058,
%T A361905 1250,1352,1458,1682,1922,2048,2178,2312,2738,2888,3042,3362,3698,
%U A361905 3872,4232,4418,4802,5000,5202,5408,5618,6050,6498,6728,6962,7442,7688,8192,8450,8978,9248,9522
%N A361905 Numbers k for which sqrt(k/2) divides k and the width at the diagonal of the symmetric representation of sigma(k) equals 1.
%C A361905 Every number in this sequence has the form 2^(2*i + 1) * k^(2*j), i, j >= 0, k >= 1.
%C A361905 The number of 1's in row a(n) of the triangle in A237048 as well as the length of that row are odd.
%F A361905 a(n) = k when A001105(n) = k and A320137(k) = 1.
%e A361905 a(4) = 32 has 4 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
%e A361905 a(5) = 50 has 5 as its single middle divisor, and its symmetric representation of sigma consists of three parts of width 1.
%e A361905 a(9) = 200 = 2^3 * 5^2 has 10 = 2 * 5 as its single middle divisor, and its symmetric representation of sigma consists of one part of maximum width 2 (A250068), but has width 1 at the diagonal.
%e A361905 a(39) = 6050 = 2^1 * 5^2 * 11^2 has 55 as its single middle divisor; it is the first number in the sequence whose symmetric representation of sigma consists of 3 parts and its central part has maximum width 2, but has width 1 at the diagonal.
%t A361905 (* Function a249223[ ] is defined in A320137 *)
%t A361905 a361905[n_] := Select[Range[n], IntegerQ[#/Sqrt[#/2]]&&Last[a249223[#]]==1&]
%t A361905 a361905[10000]
%Y A361905 Intersection of A001105 and A320137.
%Y A361905 Subsequence of A071562 and of A319796.
%Y A361905 Cf. A004171, A235791, A237048, A237270, A237271, A237593, A250068.
%K A361905 nonn
%O A361905 1,1
%A A361905 _Hartmut F. W. Hoft_, Mar 28 2023