

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.


0



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
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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



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



EXTENSIONS



STATUS

approved



