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Numbers k for which sqrt(k/2) divides k and the width at the diagonal of the symmetric representation of sigma(k) equals 1.
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%I #11 May 05 2023 19:38:03

%S 2,8,18,32,50,98,128,162,200,242,338,392,512,578,722,882,968,1058,

%T 1250,1352,1458,1682,1922,2048,2178,2312,2738,2888,3042,3362,3698,

%U 3872,4232,4418,4802,5000,5202,5408,5618,6050,6498,6728,6962,7442,7688,8192,8450,8978,9248,9522

%N 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 Every number in this sequence has the form 2^(2*i + 1) * k^(2*j), i, j >= 0, k >= 1.

%C 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 a(n) = k when A001105(n) = k and A320137(k) = 1.

%e a(4) = 32 has 4 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.

%e a(5) = 50 has 5 as its single middle divisor, and its symmetric representation of sigma consists of three parts of width 1.

%e 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 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 (* Function a249223[ ] is defined in A320137 *)

%t a361905[n_] := Select[Range[n], IntegerQ[#/Sqrt[#/2]]&&Last[a249223[#]]==1&]

%t a361905[10000]

%Y Intersection of A001105 and A320137.

%Y Subsequence of A071562 and of A319796.

%Y Cf. A004171, A235791, A237048, A237270, A237271, A237593, A250068.

%K nonn

%O 1,1

%A _Hartmut F. W. Hoft_, Mar 28 2023