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 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

%I #9 Sep 24 2018 02:40:16

%S 1,1,1,1531,97,8179,9857,6397,113,151381,7900073,320659,25684961,

%T 5963479,3414293,126665353,9138347,145606927,344182193,581101,

%U 70931614823,19274516881,4465938623,443888784679,536952085289,950571835393,1118612760917,7736464604329,962786346623

%N 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.

%C Conjecture: a(n) > 1 exists for all n > 2.

%C 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.

%C a(29) > 32*10^12. a(30) > 16*10^12. a(31) = 4149779619577. a(32) = 3853320633887. - _Donovan Johnson_, Apr 27 2008

%e 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.

%K hard,nonn

%O 0,4

%A _Alexander Adamchuk_, Sep 23 2006

%E a(20)-a(28) from _Donovan Johnson_, Apr 27 2008

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Last modified December 2 03:45 EST 2023. Contains 367505 sequences. (Running on oeis4.)