Seq sugg'd: Moving (integer) average of primes: Do terms exist for any n?

[zak seidov]

Dear SF gurus,
do terms exist for any n?
what about fini/full tags in such a case?
thanks, zak

%I A000001
%S A000001 22,2,3,3,3,3,3,5,3,5,3,3,3,5,2,3,9,3,5,4,3,3,9,5,3,3,3,3,3,3,5,3,3,3,3,2,3,11,2,3,3,33,3,3,6,2,5,3,3,9,3,3,3,2,2,3,5,3,5,3,5,3,3,3,4,3,3,4,3,3,3,5,2,3,3,10,3,5,5,3,3,3,4,3,3,15,16,3,9,3,3,3,3,32,3,2,5,3,2,3,3,2,3,3,3,3,2,3,3,2,3,3,3,5,4,3,5,2,6,5,4,3,3,3,3,3,3,15,2,8,7,3,15,3,7,4,6,3,13,3,5,3,5,3,5,3,3,3,3,3,3,5,3,3,5,3,8,10,4,2,4,3,5,2,3,3,2,5,5,3,3,3,3,3,3,3,3,3,2,3,3,3,5,2,3,6,2,3,3,7,3,4,3,2,6,3,5,5,2,3
%N A000001 a(n) = least k > 1 such that average (p(n)+...+p(n+k))/(k+1) is an integer, p(n)=nth prime.
%C A000001 Do terms a(n) exist for any n?
%e A000001 n=1, k=22 because sum(prime(i), i=1...(1+22))/23=38 integer,
n=2, k=2 because sum(prime(i), i=2...(2+2))/3=5 integer,
n=3, k=3 because sum(prime(i), i=3...(3+3))/4=9 integer,
n=7498, k=806 (largest for n<=20000) because sum(prime(i), i=7498...(7498+806))/807=80715 integer.
%O A000001 0
%K A000001 ,nonn,
%A A000001 Zak Seidov (zakseidov at yahoo.com), Jul 23 2008