A285738 Greatest prime less than 2*n^2 for n > 1, a(1) = 1.

1, 7, 17, 31, 47, 71, 97, 127, 157, 199, 241, 283, 337, 389, 449, 509, 577, 647, 719, 797, 881, 967, 1051, 1151, 1249, 1327, 1453, 1567, 1669, 1789, 1913, 2039, 2161, 2311, 2447, 2591, 2731, 2887, 3041, 3191, 3361, 3527, 3697, 3863, 4049, 4231, 4409, 4603, 4801

Offset: 1

Author

Ralf Steiner, Apr 25 2017

Keywords

• nonn
• easy

Comments

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).

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

Crossrefs

Cf. A006530, A285388, A000040 (prime numbers), A285786 (Number of primes in interval).

Programs

Mathematica

Table[Last[FactorInteger[Numerator[2^(1-2 n^2) n Binomial[2 n^2, n^2]]][[All, 1]]], {n, 1, 30}]

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

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

PARI

a(n) = if(n==1,return(1));my(i=2*n^2); while(!isprime(i), i--); i \\ David A. Corneth, Apr 26 2017

Formula

a(n) = A006530(A285388(n)).

Extensions

a(31)-a(49) from David A. Corneth, Apr 25 2017

New name from David A. Corneth, Apr 26 2017