A134775 -id:A134775 - cp's OEIS frontend

A126807 Numbers k such that A014138(k+1) (the partial sum of the first k Catalan numbers, starting 1, 2, 5, ...) is a prime.

1, 8, 10, 30, 45, 145, 794, 2772, 2787, 9796, 38288, 39191, 40856, 41202, 47379

Offset: 1

Alexander Adamchuk, Feb 23 2007

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

Cf. A014137, A121852 (sum of first k Catalan numbers A014137(k) is prime).

s[0]:=1: for n to 1000 do s[n]:= s[n-1]+binomial(2*n+2, n+1)/(n+2) end do: a:= proc (n) if isprime(s[n]) = true then n else end if end proc: seq(a(n), n= 0.. 1000); # Emeric Deutsch, Aug 28 2007

s = 0; Do[s = s + (2n)!/n!/(n+1)!; If[ PrimeQ[s], Print[n-1]], {n, 200}]

a(n) = A134775(n) - 1. - Michael S. Branicky, Jun 24 2025

a(7)-a(9) from Emeric Deutsch, Aug 28 2007
Name clarified by Jon E. Schoenfield, Mar 25 2019
a(10)-a(15) from Michael S. Branicky, Jun 25 2025

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

Original entry on oeis.org

3, 6917, 82499, 19720133460129649, 11784299926611415613401489, 3378745302877576498748105171045289001743711517992420088871061377762366601077190668071

Offset: 1

Alexander Adamchuk, Nov 11 2007

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.

			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.

Eric Weisstein's World of Mathematics, Catalan Number

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).

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 *)

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

cp's OEIS Frontend

A126807 Numbers k such that A014138(k+1) (the partial sum of the first k Catalan numbers, starting 1, 2, 5, ...) is a prime.

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