[seqfan] Re: Natural Density of Odd Squares ( Euler)

[Peter Munn]

Yes, I think Gordon has calculated the asymptotic densities of A122132 and
its complement A038838 (numbers that are divisible by the square of an odd
prime). See the relevant comments in these sequences.

Best regards,
Peter

On Tue, December 8, 2020 12:23 am, Allan Wechsler wrote:
> Gordon, perhaps you have misunderstood what it means to be a "square-free
> integer". It's not a matter of taking all the integers and discarding the
> squares. Rather, an integer is said to be "square-free" if it is not
> divisible by any square greater than 1.
>
> On Mon, Dec 7, 2020 at 1:15 PM jean-paul allouche <
> jean-paul.allouche at imj-prg.fr> wrote:
>
>> Hi
>>
>> I am not sure of what you mean by "1/( 2k-1)": is there a misprint?
>> I guess you might have meant 1/(2k-1)^2?
>> Anyway the density of squares is zero -- a fortiori the density of
>> odd or even squares is 0 as well
>>
>> best wishes
>> jean-paul
>>
>>
>> Le 07/12/2020 à 17:10, Gordon Speagle a écrit :
>> > Hello Everyone,
>> >
>> > This is my inaugural message to the group.  My name is Gordon
>> >
>> > I am not an academic mathematician, but in my investigations into
>> numbers I
>> > will sometime encounter something  that is interesting to me and I
>> find
>> > myself unsure of its role in the larger context of professional
>> > mathematics.
>> >
>> > After revisiting Euler's solution to the Basel problem and his linking
>> of
>> > pi to the distribution of square free integers, using his same logic,
>> it
>> > can be shown that the series of 1/( 2k-1) converges to  (pi2)/ 8.
>> >
>> > Heuristically, it seems that this would mean that the naturally
>> density of
>> > non odd square integers is 8/pi2.
>> >
>> > Would the density of the set of the remaining odd square  integers be
>> > 1- 8/pi2 ?
>> >
>> > It seems that follows from an Erdos theorem on natural density.
>> >
>> > I am unsure if my reasoning is sound and would like to discuss among
>> those
>> > with similar interests. Which is what precipitated my post.
>> >
>> > Thanks
>> >
>> > Gordon
>> >
>> > --
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>>
>>
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