A266241 - OEIS

%I #15 Sep 08 2022 08:46:15

%S 1,2,12,377,2504,342317

%N Integers n such that average of first n Catalan numbers (A000108) is an integer.

%C Integers n such that A014137(n-1) is divisible by n.

%C Corresponding average values are 1, 1, 6875, ...

%e 1 is a term because C(0) mod 1 = 1 mod 1 = 0.

%e 2 is a term because (C(0) + C(1)) mod 2 = (1 + 1) mod 2 = 0.

%e 12 is a term because (C(0) + C(1) + ... + C(10) + C(11)) mod 12 = (1 + 1 + ... + 16796 + 58786) mod 12 = 0.

%o (PARI) a(n) = sum(k=0, n, (2*k)!/(k!)^2/(k+1)) % (n+1);

%o for(n=0, 1e5, if(a(n)==0, print1(n+1", ")));

%o (Magma) a:=[1]; catalan:=1; sum:=catalan; for n in [2..350000] do catalan:=(catalan*(4*n-6)) div n; sum+:=catalan; if (sum mod n) eq 0 then a[#a+1]:=n; end if; end for; a; // _Jon E. Schoenfield_, Dec 25 2015

%Y Cf. A000108, A014137.

%K nonn,more

%O 1,2

%A _Altug Alkan_, Dec 25 2015

%E a(6) from _Jon E. Schoenfield_, Dec 25 2015