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

[Tomasz Ordowski]

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
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https://oeis.org/A276710