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A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.

%I A365265 #27 Aug 30 2023 20:50:14
%S A365265 2,8,18,32,128,162,200,392,512,882,968,1352,1458,2048,2178,3042,3872,
%T A365265 5000,5202,5408,6498,8192,9248,9522,11552,13122,15138,16928,17298,
%U A365265 19208,26912,30752,32768,36992,43218,43808,46208,53792,58482,59168,67712,70688
%N A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.
%C A365265 Every number a(n) has the form 2^(2*i + 1) * s^2, i>= 0 and s odd, the single middle divisor of a(n) is sqrt(a(n)/2), and sqrt(2*a(n)) - 1 = floor((sqrt(8*n + 1) - 1)/2) = A003056(a(n)).
%C A365265 The least number in the sequence with 3 odd prime divisors is a(126) = 1630818 = 2^1 * 3^2 * 7^2 * 43^2.
%C A365265 Conjecture: Let a(n) = 2^(2i+1) * s^2, i>=0 and s odd, be a number in the sequence.
%C A365265 (1) For any odd prime divisor p of s, number a(n) * p^2 is in the sequence.
%C A365265 (2) For any odd prime p not a divisor of s, number a(n) * p^2 is in the sequence if p satisfies sqrt(2*a(n)) < p < 2*a(n).
%e A365265 a(5) = 128 = 2^7  has 2^3 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
%e A365265 a(10) = 882 = 2 * 3^2 * 7^2 has 3 * 7 as its single middle divisor, its symmetric representation of sigma is the smallest in this sequence of maximum width 3, consists of one part, and has width 1 at the diagonal.
%e A365265 A table of ranges for the single odd prime factor p for numbers k in the sequence having the form 2^(2i+1) * p^(2j), i>=0 and j>0, indexed by exponent 2i+1 of 2 in number k. The lower bound is A014210(i+1) and the upper bound is A014234(2(i+1)) = A104089(i+1):
%e A365265 ---------------------
%e A365265   2i+1  /---- p ----/
%e A365265 ---------------------
%e A365265   1       3  ..    3
%e A365265   3       5  ..   13
%e A365265   5      11  ..   61
%e A365265   7      17  ..  251
%e A365265   9      37  .. 1021
%e A365265 ...
%t A365265 (* a2[ ] and its support functions are defined in A249223 *)
%t A365265 a365265Q[n_] := Module[{list=If[Divisible[n, Sqrt[n/2]], a2[n], {0}]}, Last[list]==1&&AllTrue[list, #>0&]]
%t A365265 a365265[{m_, n_}] := Select[Range[m, n], a365265Q]
%t A365265 a365265[{1,75000}]
%Y A365265 Intersection of A361903 and A361905.
%Y A365265 Also subsequence of the following sequences: A001105, A071562, A238443 = A174973, A319796, A320137.
%Y A365265 The powers of 2 with an odd index (A004171) form a subsequence.
%Y A365265 Cf. A003056, A014210, A014234, A104089, A235791, A237048, A237270, A237271, A237593, A249223, A250068.
%K A365265 nonn
%O A365265 1,1
%A A365265 _Hartmut F. W. Hoft_, Aug 29 2023