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%I #56 Sep 08 2022 08:46:18
%S 3,5,7,9,10,11,13,14,15,17,18,19,21,22,23,25,26,27,29,30,31,33,34,35,
%T 36,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,54,55,57,58,59,60,61,
%U 62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,81,82,83,84,85,86,87,89,90,91,92,93,94,95,97,98,99,100,101,102,103,105
%N Numbers n having at least one odd divisor greater than sqrt(2*n).
%C Conjecture 1: also numbers n such that the symmetric representation of sigma(n) has at least one pair of equidistant subparts.
%C Conjecture 2: the number of pairs of equidistant subparts in the symmetric representation of sigma(k) equals the number of odd divisors of k greater than sqrt(2*k), with k >= 1.
%C For more information about the subparts see A279387.
%H Indranil Ghosh, <a href="/A281005/b281005.txt">Table of n, a(n) for n = 1..1000</a>
%e 18 is in the sequence because one of its odd divisors is 9, and 9 is greater than 6, the square root of 2*18.
%e On the other hand the symmetric representation of sigma(18) has only one part of size 39, which is formed by a central subpart of size 35 and a pair of equidistant subparts [2, 2]. Since there is at least one pair of equidistant subparts, so 18 is in the sequence.
%e From _Omar E. Pol_, Dec 18 2020: (Start)
%e The 17th row of triangle A237593 is [9, 4, 2, 1, 1, 1, 1, 2, 4, 9] and the 18th row of the same triangle is [10, 3, 2, 2, 1, 1, 2, 2, 3, 10], so the diagram of the symmetric representation of sigma(18) = 39 is constructed as shown below in figure 1:
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%e . _ _| _| _ _| _| 2
%e . | | 39 | _ _|
%e . | _ _| | |_ _|
%e . | | | | 2
%e . _ _ _ _ _ _ _ _ _| | _ _ _ _ _ _ _ _ _| |
%e . |_ _ _ _ _ _ _ _ _ _| |_ _ _ _ _ _ _ _ _ _|
%e . 35
%e .
%e . Figure 1. The symmetric Figure 2. After the dissection
%e . representation of sigma(18) of the symmetric representation
%e . has one part of size 39. of sigma(18) into layers of
%e . width 1 we can see three subparts.
%e . The first layer has one subpart of
%e . size 35. The second layer has
%e . two equidistant subparts of size 2,
%e . so 18 is in the sequence.
%e (End)
%t Select[Range@ 120, Count[Divisors@ #, d_ /; And[OddQ@ d, d > Sqrt[2 #]]] > 0 &] (* _Michael De Vlieger_, Feb 07 2017 *)
%o (PARI) isok(n) = my(s=sqrt(2*n)); sumdiv(n, d, (d % 2) && (d > s)) > 0; \\ _Michel Marcus_, Jan 15 2020
%o (Magma) [k:k in [1..110] | not forall{d:d in Divisors(k)| IsEven(d) or d le Sqrt(2*k)}]; // _Marius A. Burtea_, Jan 15 2020
%Y Complement of A082662.
%Y Indices of positive terms in A131576.
%Y Cf. A000203, A001227, A067742, A071561, A196020, A235791, A236104, A237048, A237270, A237271, A237571, A237593, A244050, A245092, A249351, A262626, A279387, A280850, A280851, A296508.
%K nonn
%O 1,1
%A _Omar E. Pol_, Feb 06 2017
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