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A217772
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a(n) = ((p+1)*(3p)!/((2p-1)!*(p+1)!*2p) - 3)/(3p^3), where p is the n-th prime.
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2
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1, 8, 113, 48469, 1232351, 1002175798, 30956114561, 32956274508457, 1386101220044940571, 50017672586399947073, 2548160990547719392420658, 3694160975065765801289789930, 142486973648670437070915061157
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OFFSET
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2,2
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COMMENTS
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This sequence is based on Gary Detlefs's conjecture, which he posted as a comment to A005809. His conjecture is equivalent to the conjecture that the Diophantine equation ((n+1)*(3*n)!/((2*n-1)!*(n+1)!*2*n)-3)/n^3 = m has integer solutions m for all odd primes n.
Additionally I conjecture that all m are divisible by 3, therefore terms of this sequence a(n) = m/3.
It is also notable that for quite a few values of n (2, 3, 4, 5, 6, 7, 17, 19, 21, 22, 23, 24, 25, 26, 35, 39, 43, ...) a(n+1) = a(n) mod 7.
The values of this sequence's terms are replicated by conjectured general formula, given in A223886 (and also added to the formula section here) for k=3, j=1 and n>=2. - Alexander R. Povolotsky, Apr 18 2013
For n>=3 and k>=2 ((binomial(k*p,p)-k)/p^3)/k is an integer. For k=2 this is the Wolstenholme quotient (A034602) and for k=3 the current sequence. - Peter Luschny, Feb 09 2016
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LINKS
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FORMULA
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a(n) = ((binomial (j*k*prime(n), j*prime(n)) - binomial(k*j, j))/(k*prime(n)^3) for k=3, j=1 and n>=2 (conjectured). - Alexander R. Povolotsky, Apr 18 2013
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MAPLE
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WQ := proc(n, k) local p; p := ithprime(n); ((binomial(k*p, p)-k)/p^3)/k end:
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PROG
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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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