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%I #49 Sep 08 2022 08:44:52
%S 1,5,265,2367,237493,2576561,338350897,616410400171,7811559753873,
%T 17236200860123055,3081677433937346539,41741941495866750557,
%U 7829195555633964779233,21066131970056662377432067,59296957594629000880904587621,844326030443651782154010715715
%N Wolstenholme quotient W_p = (binomial(2p-1,p) - 1)/p^3 for prime p=A000040(n).
%C Equivalently, (binomial(2p,p)-2)/(2*p^3) where p runs through the primes >=5.
%C 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
%D R. K. Guy, Unsolved Problems in Number Theory, Sect. B31.
%H Robert Israel, <a href="/A034602/b034602.txt">Table of n, a(n) for n = 3..263</a>
%H 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:<a href="http://doi.org/10.1007/s10958-018-3948-0">10.1007/s10958-018-3948-0</a> arXiv:<a href="http://arxiv.org/abs/1602.02632">1602.02632</a>
%H R. J. McIntosh, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7144.pdf">On the converse of Wolstenholme's theorem</a>, Acta Arithmetica 71:4 (1995), 381-389.
%H R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%H J. Sondow, Extending Babbage's (non-)primality tests, in <a href="https://doi.org/10.1007/978-3-319-68032-3_19">Combinatorial and Additive Number Theory II</a>, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; <a href="http://arxiv.org/abs/1812.07650">arXiv:1812.07650 [math.NT]</a>, 2018.
%F a(n) = (A088218(p)-1)/p^3 = (A001700(p-1)-1)/p^3 = (A000984(p)-2)/(2*p^3), where p=A000040(n).
%F a(n) = A087754(n) / 2.
%F 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
%F a(n) = A263882(n)/prime(n) for n > 2. - _Jonathan Sondow_, Nov 23 2015
%e Binomial(10,5)-2 = 250; 5^3=125 hence a(5)=1.
%p f:= proc(n) local p;
%p p:= ithprime(n);
%p (binomial(2*p-1,p)-1)/p^3
%p end proc:
%p map(f, [$3..30]); # _Robert Israel_, Dec 19 2018
%t Table[(Binomial[2 Prime[n] - 1, Prime[n] - 1] - 1)/Prime[n]^3, {n, 3, 20}] (* _Vincenzo Librandi_, Nov 23 2015 *)
%o (Magma) [(Binomial(2*p-1,p)-1) div p^3: p in PrimesInInterval(4,100)]; // _Vincenzo Librandi_, Nov 23 2015
%Y Cf. A177783 (alternative definition of Wolstenholme quotient), A072984, A092101, A092103, A092193, A128673, A217772, A223886, A263882.
%Y Cf. A268512, A268589, A268590.
%K nonn
%O 3,2
%A _N. J. A. Sloane_.
%E Edited by _Max Alekseyev_, May 14 2010
%E More terms from _Vincenzo Librandi_, Nov 23 2015
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