A134776 Primes that are the sum of first k Catalan numbers for some integer k.

3, 6917, 82499, 19720133460129649, 11784299926611415613401489, 3378745302877576498748105171045289001743711517992420088871061377762366601077190668071

Offset: 1

Author

Alexander Adamchuk, Nov 11 2007

Keywords

• hard
• nonn

Comments

Next term a(7) has 475 decimal digits and is too large to include. Corresponding numbers k such that the sum of first k Catalan numbers is a prime are listed in A134775.

Examples

			a(1) = 3 because C(1) + C(2) = 1 + 2 = 3 is a prime.
a(2) = 6917 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. A134775 (numbers k such that the sum of first k Catalan numbers is a prime).

Cf. A014138 (partial sums of Catalan numbers).

Cf. A000108 (Catalan numbers).

Programs

Mathematica

f=0; Do[ f = f + Binomial[ 2n, n ]/(n+1); If[ PrimeQ[f], Print[ {n, f} ] ], {n, 1, 1000} ]
Select[Accumulate[CatalanNumber[Range[200]]],PrimeQ] (* Harvey P. Dale, Mar 01 2019 *)

Formula

a(n) = A014138( A134775(n) - 1 ).