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A223886
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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.
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2
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OFFSET
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2,1
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COMMENTS
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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.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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