login
A281005
Numbers n having at least one odd divisor greater than sqrt(2*n).
7
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
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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 06 2017
STATUS
approved