OFFSET
0,4
COMMENTS
Conjecture: a(n) > 1 exists for all n > 2.
a(0) = a(1) = a(2) = 1 because Sum_{i=1..8} prime(i)^2 = 1027 = 13*79 is odd and composite and (Sum_{i=k..k+7} prime(i)^2)/2^n is always even (composite) for k > 1 and n = {0,1,2}. The sum of the squares of any 8 odd primes is always divisible by 8 because, for i > 2, prime(i)^2 mod 24 = 1 and, for i=2, prime(i)^2 mod 8 = 3^2 mod 8 = 1. Thus the sum of the squares of any 8 odd primes is of the form 8*k.
a(29) > 32*10^12. a(30) > 16*10^12. a(31) = 4149779619577. a(32) = 3853320633887. - Donovan Johnson, Apr 27 2008
EXAMPLE
a(4) = 97 because Sum_{i=k..k+7} prime(i)^2 = 2^4*97 for k = 2 and 2^4 does not divide Sum_{i=1..8} prime(i)^2 = 13*79.
CROSSREFS
KEYWORD
hard,nonn
AUTHOR
Alexander Adamchuk, Sep 23 2006
EXTENSIONS
a(20)-a(28) from Donovan Johnson, Apr 27 2008
STATUS
approved