2*(A056855(n))/(phi(n)*n)

[Benoit Cloitre]

It is true from Leudesdorf theorem page 101 of Hardy & Wright fifht 
edition (the last line) . If I'm not wrong more is true :

12 *A056855(n)/(phi(n)*core(n)*n) is an integer where 
core(n)=squarefree part of n.

12 *A056855(n)/n^2 is an integer

It seems interesting to have more properties of those numbers 
S(n)=sum(1/k, 1<=k<=n, (k,n)=1)

writing :

P(n)=prod(k, 1<=k<=n, (k,n)=1)  (A001783)

(i) one has denominator of S(n) divides P(n) for any n  and denominator 
of S(n)=P(n) iff n is a very round number (A048597)

(ii) S(n) and Harmonic numbers H(n)=sum(1/k,1<=k<=n) are related by 
this property :

n doesn't divide the {denominator of H(n)-Q(n)}  if and only if   
{numerator of  n*(H(n)-1)} - {numerator of H(n)} + {denominator of 
H(n)} >0

Those n's are given by  A074791 : 6,18,20,21,33,42,54,63,66,77,...

Benoit Cloitre



>> 2 * (A056855(n)) /(phi(n)*n)
>> ...
>> 2, 1, 1, 1, 5, 1, 84, 11, ...
>> (And this sequence is not in the EIS, but I am submitting the few 
>> terms
>> above.)
>
>     Here's more.
>
> %S Axxxxxx 2,1,1,1,5,1,84,11,184,15,193248,23,19056960,833,33740,64035,
> %T Axxxxxx 520105017600,2473,130859579289600,203685,963513600,23748417,
> %U Axxxxxx 16397141420298240000,645119,555804546402631680,8527366575
>
>
>> I asked this question years ago, but is
>> 2 * (A056855(n)) /(phi(n)*n)
>> always an integer?
>>
>> I conjecture it is.
>
>     Well, the first 5000 terms are integers.
>     And the conjecture is true whenever N and phi(N)/2 are coprime.
>     But that leaves much in doubt.
>
> --
> Don Reble  djr at nk.ca
>
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