A281005 Numbers n having at least one odd divisor greater than sqrt(2*n).

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 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

Offset

1,1

Comments

Conjecture 1: also numbers n such that the symmetric representation of sigma(n) has at least one pair of equidistant subparts.

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.

For more information about the subparts see A279387.

Links

Indranil Ghosh, Table of n, a(n) for n = 1..1000

Example

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.
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.
From Omar E. Pol, Dec 18 2020: (Start)
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:
.                                     _                                      _
.                                    | |                                    | |
.                                    | |                                    | |
._                                   | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                             _ _ _ _| |                             _ _ _ _| |
.                            |    _ _ _|                            |  _ _ _ _|
.                           _|   |                                 _| | |
.                         _|  _ _|                               _|  _|_|
.                     _ _|  _|                               _ _|  _|    2
.                    |     |  39                            |  _ _|
.                    |  _ _|                                | |_ _|
.                    | |                                    | |    2
.   _ _ _ _ _ _ _ _ _| |                   _ _ _ _ _ _ _ _ _| |
.  |_ _ _ _ _ _ _ _ _ _|                  |_ _ _ _ _ _ _ _ _ _|
.                                                              35
.
.   Figure 1. The symmetric               Figure 2. After the dissection
.   representation of sigma(18)           of the symmetric representation
.   has one part of size 39.              of sigma(18) into layers of
.                                         width 1 we can see three subparts.
.                                         The first layer has one subpart of
.                                         size 35. The second layer has
.                                         two equidistant subparts of size 2,
.                                         so 18 is in the sequence.
(End)

Mathematica

Select[Range@ 120, Count[Divisors@ #, d_ /; And[OddQ@ d, d > Sqrt[2 #]]] > 0 &] (* Michael De Vlieger, Feb 07 2017 *)

Prog

(PARI) isok(n) = my(s=sqrt(2*n)); sumdiv(n, d, (d % 2) && (d > s)) > 0; \\ Michel Marcus, Jan 15 2020
(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

Crossrefs

Complement of A082662.

Indices of positive terms in A131576.

Cf. A000203, A001227, A067742, A071561, A196020, A235791, A236104, A237048, A237270, A237271, A237571, A237593, A244050, A245092, A249351, A262626, A279387, A280850, A280851, A296508.

Sequence in context: A096849 A369361 A326980 * A080259 A067715 A344291

Adjacent sequences: A281002 A281003 A281004 * A281006 A281007 A281008

Keyword

nonn

Author

Omar E. Pol, Feb 06 2017

Status

approved