[seqfan] Re: integers that are either square or thrice a square, where do such beasts hide?

[Wouter Meeussen]

Further scrutiny broadens this to integers of the form  (1 or a square)*(1 
or 3)*(1 or prime).
Least integer not of this form is 10.
List up to 100 :
1,2,3,4,5,6,7,8,9,11,12,13,15,16,17,18,19,20,21,23,24,25,27,28,29,31,32,33,36,37,39,41,43,44,45,47,48,49,50,
51,52,53,54,57,59,60,61,63,64,67,68,69,71,72,73,75,76,79,80,81,83,84,87,89,92,93,96,97,98,99,100

Integers *not* in scope are maybe OEIS-worthy:
10,14,22,26,30,34,35,38,40,42,46,55,56,58,62,65,66,70,74,77,78,82,85,86,88,90,91,94,95

For the first 100 integers, 71% are in-scope, for the first 10^3, still 49%
for the first 10^4, 35% and for the first 10^6 only 20%.
Between 9 10^5 and 10^6 it has dwindled to 18.5%.

I encountered this sequence when comparing a bunch of Singular Binary 
Bracelets: size p=2q+1 with p and q prime, q unit beads, resultant amplitude 
squared is integer.
There is a link to Algebraic Number Fields, but I'm still groping to grok 
it.

warning: Mathematica code below this line!
-------------------------------
Select[Range[300], ! MatchQ[ DeleteCases[ FactorInteger[#], {3, _} | {_, 
_?EvenQ}], {} | {{_, _?OddQ}}] &]
-------------------------------

Wouter.

-----Original Message----- 
From: Allan Wechsler
Sent: Wednesday, August 29, 2018 5:46 PM
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: integers that are either square or thrice a square, 
where do such beasts hide?

There is a near-hit at A230781. The first difference is that your sequence
contains 49.

In general, A230781 excludes from your sequence numbers of the form x^2 +
xy + y^2, for 0 < x < y; that is, numbers in A024606.

If this really arises naturally, it seems worth submitting.

On Wed, Aug 29, 2018 at 11:02 AM Wouter Meeussen 
<wouter.meeussen at telenet.be>
wrote:

> do they occur anywhere naturally? in combinatorics or in number theory?
> I bumped into them by accident looking at bracelet-stuff.
>
> Wouter.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

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