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A034602
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Wolstenholme quotient W_p = (binomial(2p-1,p) - 1)/p^3 for prime p=A000040(n).
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31
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1, 5, 265, 2367, 237493, 2576561, 338350897, 616410400171, 7811559753873, 17236200860123055, 3081677433937346539, 41741941495866750557, 7829195555633964779233, 21066131970056662377432067, 59296957594629000880904587621, 844326030443651782154010715715
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OFFSET
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3,2
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COMMENTS
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Equivalently, (binomial(2p,p)-2)/(2*p^3) where p runs through the primes >=5.
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=2, j=1 and n>=3. - Alexander R. Povolotsky, Apr 18 2013
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, Sect. B31.
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LINKS
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R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0 arXiv:1602.02632
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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=2, j=1, and n>=3. - Alexander R. Povolotsky, Apr 18 2013
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EXAMPLE
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Binomial(10,5)-2 = 250; 5^3=125 hence a(5)=1.
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MAPLE
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f:= proc(n) local p;
p:= ithprime(n);
(binomial(2*p-1, p)-1)/p^3
end proc:
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MATHEMATICA
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Table[(Binomial[2 Prime[n] - 1, Prime[n] - 1] - 1)/Prime[n]^3, {n, 3, 20}] (* Vincenzo Librandi, Nov 23 2015 *)
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PROG
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(Magma) [(Binomial(2*p-1, p)-1) div p^3: p in PrimesInInterval(4, 100)]; // Vincenzo Librandi, Nov 23 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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