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%I #68 May 01 2017 09:39:22
%S 1,7,17,31,47,71,97,127,157,199,241,283,337,389,449,509,577,647,719,
%T 797,881,967,1051,1151,1249,1327,1453,1567,1669,1789,1913,2039,2161,
%U 2311,2447,2591,2731,2887,3041,3191,3361,3527,3697,3863,4049,4231,4409,4603,4801
%N Greatest prime less than 2*n^2 for n > 1, a(1) = 1.
%C a(n) for n>1 is prime. Further the upper part of at least n in i well-ordered prime factors p_i(n) of the numerator of 2^(1-2 n^2) n binomial(2 n^2, n^2) (A285388(n)) consists of only single factors which form especially a complete part of the prime numbers p with 3 < 2(n-1)^2 < p <= a(n) < 2n^2. Thus the complete union of {2,3} and {p_i(m)} for m from 2 to n gives all prime numbers p <= a(n).
%C Alternative definitions are "Greatest prime factor of the numerator of 2^(1-2 n^2) n binomial(2 n^2, n^2)". and "Greatest prime factor of numerator of sum{k=0..n^2-1}(binomial(2k,k)/4^k)/n". - _David A. Corneth_, Apr 26 2017
%F a(n) = A006530(A285388(n)).
%t Table[Last[FactorInteger[Numerator[Table[2^(1-2 n^2) n Binomial[2 n^2, n^2], {n, 1, 30}]][[k]][[All, 1]]], {k, 1, 30}]
%o (PARI) a(n) = my(f = factor(sum(k = 0, n^2-1, (binomial(2*k, k)/4^k))/n)[, 1]); f[#f] \\ _David A. Corneth_, Apr 25 2017
%o (PARI) a(n) = if(n==1,1,my(f=factor(n*binomial(2*n^2, n^2))[,1]); f[#f]) \\ _David A. Corneth_, Apr 26 2017
%o (PARI) a(n) = if(n==1,return(1));my(i=2*n^2); while(!isprime(i), i--); i \\ _David A. Corneth_, Apr 26 2017
%Y Cf. A006530, A285388, A000040 (prime numbers), A285786 (Number of primes in interval).
%K nonn,easy
%O 1,2
%A _Ralf Steiner_, Apr 25 2017
%E a(31)-a(49) from _David A. Corneth_, Apr 25 2017
%E New name from _David A. Corneth_, Apr 26 2017
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