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A263882 Babbage quotients b_p = (binomial(2p-1, p-1) - 1)/p^2 with p = prime(n).

%I A263882 #45 Sep 08 2022 08:46:14
%S A263882 1,5,35,2915,30771,4037381,48954659,7782070631,17875901604959,
%T A263882 242158352370063,637739431824553035,126348774791431208099,
%U A263882 1794903484322270273951,367972191114796344623951,1116504994413003106003899551,3498520498083111051973370669639
%N A263882 Babbage quotients b_p = (binomial(2p-1, p-1) - 1)/p^2 with p = prime(n).
%C A263882 Charles Babbage proved in 1819 that b_p is an integer for prime p > 2. In 1862 Wolstenholme proved that the Wolstenholme quotient W_p = b_p / p is an integer for prime p > 3; see A034602.
%C A263882 The quotient b_n is an integer for composite n in A267824. No composite n is known for which W_n is an integer.
%D A263882 R. K. Guy, Unsolved Problems in Number Theory, Sect. B31.
%H A263882 Robert Israel, <a href="/A263882/b263882.txt">Table of n, a(n) for n = 2..260</a>
%H A263882 C. Babbage, <a href="http://books.google.com/books?id=KrA-AAAAYAAJ&amp;pg=PA46">Demonstration of a theorem relating to prime numbers</a>, Edinburgh Philosophical Journal, 1 (1819), 46-49.
%H A263882 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 A263882 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.
%H A263882 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem">Wolstenholme's theorem</a>
%H A263882 J. Wolstenholme, <a href="http://books.google.com/books?id=vL0KAAAAIAAJ&amp;pg=PA35">On certain properties of prime numbers</a>, Quarterly Journal of Pure and Applied Mathematics, 5 (1862), 35-39.
%F A263882 a(n) = prime(n)*A034602(n) for n > 2.
%F A263882 a(PrimePi(A088164(n))) == 0 mod A088164(n)^2.
%e A263882 a(2) = (binomial(2*3-1,3-1) - 1)/3^2 = (binomial(5,2) - 1)/9 = (10-1)/9 = 1.
%p A263882 map(p -> (binomial(2*p-1,p-1)-1)/p^2, select(isprime,[seq(i,i=3..100,2)])); # _Robert Israel_, Nov 24 2015
%t A263882 Table[(Binomial[2*Prime[n] - 1, Prime[n] - 1] - 1)/Prime[n]^2, {n, 2, 17}]
%t A263882 Table[(Binomial[2p-1,p-1]-1)/p^2,{p,Prime[Range[2,20]]}] (* _Harvey P. Dale_, Jul 20 2019 *)
%o A263882 (Magma) [(Binomial(2*NthPrime(n)-1, NthPrime(n)-1)-1)/NthPrime(n)^2: n in [2..20]]; // _Vincenzo Librandi_, Nov 25 2015
%Y A263882 Cf. A034602, A088164, A099906, A267824.
%K A263882 nonn
%O A263882 2,2
%A A263882 _Jonathan Sondow_, Nov 22 2015