A223886 Numbers (binomial(j*k*prime(n), j*prime(n)) - binomial(k*j, j))/(k*prime(n)^3) for k=4, j=3 and n>=2.

871695, 106388178385, 23847838715080655, 2591856748839247419391825095, 1049841259371423735816549330164685, 216822871259048720341882553570648156557191421

Offset: 2

Author

Alexander R. Povolotsky, Mar 28 2013

Keywords

• nonn

Comments

This sequence (together with already present in the OEIS A034602 and A217772) is based on Gary Detlefs' conjecture, which he disclosed to me in a private communication on 3/29/13 and recently he gave me permission to make it public. Specifically he wrote to me the following: "I have a conjecture which is broader than the one I submitted, having to do with binomial(k*n,n) mod n^3. It appears that binomial(j*k*n,j*n) mod n^3 will be binomial(k*j,j) for n sufficiently large."

In effect above conjecture further extends Wolstenholme's and Ljunggren's ideas and could also be expressed as follows: starting with some specific (for any given unchanged values of integers k>0 and j>0) sufficiently large value of n=N and further on for n>N it is true that (binomial(j*k*prime(n), j*prime(n)) - binomial(k*j, j))/k/(prime(n))^3 = m(j, k, n ), where m(j, k, n ) are integer values.

Note that the values of A034602 are replicated by above general formula for k=2, j=1 and n>=3 and the values of A217772 are replicated by the same formula for k=3, j=1 and n>=2.

Links

• R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv preprint arXiv:1111.3057 [math.NT], 2011.

Crossrefs

Cf. A034602, A217772.