login
a(n) = numerator((2*n-1)^2/(2*(2*n)!)).
7

%I #55 Feb 26 2024 15:44:23

%S 1,1,3,5,7,1,11,13,1,17,19,1,23,1,1,29,31,1,1,37,1,41,43,1,47,1,1,53,

%T 1,1,59,61,1,1,67,1,71,73,1,1,79,1,83,1,1,89,1,1,1,97,1,101,103,1,107,

%U 109,1,113,1,1,1,1,1,1,127

%N a(n) = numerator((2*n-1)^2/(2*(2*n)!)).

%C 1's between primes correspond to odd nonprimes (see A047846).

%H Reinhard Zumkeller, <a href="/A128059/b128059.txt">Table of n, a(n) for n = 0..1000</a>

%F Conjecture: a(n) = denominator(f(n-1)) with f(n) = lcm(2,3,4,5,...,n)*(Sum_{k=0..n} frac(Bernoulli(2*k))*binomial(n+k,k)). - _Yalcin Aktar_, Jul 23 2008

%F a(n) = 2*n-3 if 2*n-3 is prime and a(n) = 1 otherwise. a(n+4) = A145737(n+2), for n >= 1. - _Artur Jasinski_, Nov 29 2008

%F a(n+1) = denominator( (2n)!/(2n+1) ), n > 0. - _Wesley Ivan Hurt_, Jun 19 2013

%F a(n+1) = abs(2n*(pi(2n) - pi(2n-2)) - 1) where abs is the absolute value function and pi is the prime counting function (A000720). - _Anthony Browne_, Jun 28 2016

%F a(n+1) = denominator(Bernoulli(2*n)*(2*n)!) = numerator(Clausen(2*n,1)/(2*n)!) with Clausen defined in A160014. - _Peter Luschny_, Sep 25 2016

%p A128059 := proc(n): numer(((2*n-1)^2)/(2*(2*n)!)) end: seq(A128059(n), n=0..64); # _Artur Jasinski_, Nov 29 2008

%p A128059 := proc(n): if isprime(2*n-1) then 2*n-1 else 1 fi: end: seq(A128059(n), n=0..64); # _Johannes W. Meijer_, Oct 25 2012, Jun 01 2016

%t Table[Numerator[(2 n - 1)^2/(2 (2 n)!)], {n, 0, 64}] (* _Michael De Vlieger_, Jun 01 2016 *)

%o (Haskell)

%o a128059 0 = 1

%o a128059 n = f n n where

%o f 1 _ = 1

%o f x q = if a010051' q' == 1 then q' else f x' q'

%o where x' = x - 1; q' = q + x'

%o -- _Reinhard Zumkeller_, Jun 14 2015

%o (Python)

%o from sympy import isprime

%o def A128059(n): return a if isprime(a:=(n<<1)-1) else 1 # _Chai Wah Wu_, Feb 26 2024

%Y Cf. A010051, A128059, A145737.

%Y Essentially the odd bisection of A089026.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Feb 13 2007