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.

2, 8, 18, 32, 50, 98, 128, 162, 200, 242, 338, 392, 512, 578, 722, 882, 968, 1058, 1250, 1352, 1458, 1682, 1922, 2048, 2178, 2312, 2738, 2888, 3042, 3362, 3698, 3872, 4232, 4418, 4802, 5000, 5202, 5408, 5618, 6050, 6498, 6728, 6962, 7442, 7688, 8192, 8450, 8978, 9248, 9522

Offset: 1

Author

Hartmut F. W. Hoft, Mar 28 2023

Keywords

• nonn

Comments

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

The number of 1's in row a(n) of the triangle in A237048 as well as the length of that row are odd.

Examples

			a(4) = 32 has 4 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(5) = 50 has 5 as its single middle divisor, and its symmetric representation of sigma consists of three parts of width 1.
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.
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.
		

Crossrefs

Intersection of A001105 and A320137.

Subsequence of A071562 and of A319796.

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

Programs

Mathematica

(* Function a249223[ ] is defined in A320137 *)
a361905[n_] := Select[Range[n], IntegerQ[#/Sqrt[#/2]]&&Last[a249223[#]]==1&]
a361905[10000]

Formula

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