A134775 Numbers k such that the sum of first k Catalan numbers is a prime.

2, 9, 11, 31, 46, 146, 795, 2773, 2788, 9797, 38289, 39192, 40857, 41203, 47380

Offset: 1

Author

Alexander Adamchuk, Nov 11 2007

Keywords

• hard
• more
• nonn

Comments

Corresponding primes are listed in A134776. A134776(n) = A014138(a(n)-1).

a(16) > 156000. - Michael S. Branicky, Jun 26 2025

Examples

			a(1) = 2 because C(1) + C(2) = 1 + 2 = 3 is a prime.
a(2) = 9 because C(1) + C(2) + C(3) + C(4) + C(5) + C(6) + C(7) + C(8) + C(9) = 1 + 2 + 5 + 14 + 42 + 132 + 429 + 1430 + 4862 = 6917 is a prime.
		

Links

• Eric Weisstein's World of Mathematics, Catalan Number

Crossrefs

Cf. A134776 (primes that are the sum of first n Catalan numbers).

Cf. A014138 (partial sums of Catalan numbers).

Cf. A000108 (Catalan numbers).

Cf. A126807.

Programs

Maple

for n to 3000 do c[n]:= binomial(2*n,n)/(n+1) end do: a:=proc(n) if isprime(add(c[j],j=1..n))=true then n else end if end proc: seq(a(n),n=1..3000); # Emeric Deutsch, Nov 19 2007

Mathematica

f=0; Do[ f = f + Binomial[ 2n, n ]/(n+1); If[ PrimeQ[f], Print[ {n, f} ] ], {n, 1, 1000} ]
Flatten[Position[Accumulate[CatalanNumber[Range[1000]]],?PrimeQ]] (* _Harvey P. Dale, Jan 28 2013 *)

Formula

a(n) = A126807(n) + 1. - Michael S. Branicky, Jun 24 2025

Extensions

a(8)-a(9) from Emeric Deutsch, Nov 19 2007

a(10) from Ryan Propper, Jan 06 2008

a(11)-a(15) from Michael S. Branicky, Jun 25 2025