[seqfan] Re: On sums and products of the fractions C(n,k)/n

[Tomasz Ordowski]

P.S. The numbers I asked about are a subset of A276710:
https://oeis.org/A276710
Pseudoprimes A001567 in this sequence are
1105, 1729, 2465, 2821, 6601, 11305, 13981, 15841, ...
Note that 11305 and 13981 are not Carmichael numbers.
Even pseudoprimes A006935 extend beyond the database A276710:
https://oeis.org/A276710/b276710.txt


pon., 24 maj 2021 o 14:38 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> Dear readers!
>
> Let C(n,k) = binomial(n,k).
> As is well known, n|C(n,k) if and only if gcd(k,n)=1.
> It should be noted that Sum_{0<k<n} C(n,k) = 2^n-2.
> If n is an odd or even Fermat pseudoprime to base 2,
> then Sum_{0<k<n,gcd(k,n)>1} C(n,k)/n is an integer.
> Most importantly, no fraction in this sum is an integer.
> Question: are there composite numbers n such that
> Product_{0<k<n,gcd(k,n)>1} C(n,k)/n is an integer?
> If so, are there such (odd or even) pseudoprimes?
> Also in this product, no fraction is an integer.
>
> Best regards,
>
> Thomas
> _____________
> https://oeis.org/A276710
>
>