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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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%I #15 Mar 26 2019 19:27:20

%S 1,8,10,30,45,145,794,2772,2787

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

%p 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

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

%Y Cf. A014138, A000108.

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

%K more,nonn

%O 1,2

%A _Alexander Adamchuk_, Feb 23 2007

%E 3 more terms from _Emeric Deutsch_, Aug 28 2007

%E Name clarified by _Jon E. Schoenfield_, Mar 25 2019