A122707 Smallest prime equal to the sum of squares of 8 consecutive primes divided by 2^n, or 1 if such a prime does not exist.

1, 1, 1, 1531, 97, 8179, 9857, 6397, 113, 151381, 7900073, 320659, 25684961, 5963479, 3414293, 126665353, 9138347, 145606927, 344182193, 581101, 70931614823, 19274516881, 4465938623, 443888784679, 536952085289, 950571835393, 1118612760917, 7736464604329, 962786346623

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

Links

Table of n, a(n) for n=0..28.

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

Sequence in context: A252414 A333721 A030061 * A347200 A057327 A057328

Adjacent sequences: A122704 A122705 A122706 * A122708 A122709 A122710

Keyword

hard,nonn

Author

Alexander Adamchuk, Sep 23 2006

Extensions

a(20)-a(28) from Donovan Johnson, Apr 27 2008

Status

approved