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%I #8 Mar 31 2012 10:23:48
%S 1,3,5,6,7,7,9,8,9,12,7,9,4,4,8,11,6,11,7,8,14,7,8,11,6,10,9,8,8,11,6,
%T 10,13,7,6,9,10,8,8,10,5,10,15,6,11,9,14,7,8,16,12,10,5,10,9,8,10,8,7,
%U 10,11,13,12,6,12,9,4,10,12,13,8,14,7,2,13,7
%N a(n) is the number of primes in the form of 4^n+/-4^k+/-1, while 0 <= k < n.
%C Conjecture: all elements of this sequence is greater than 0.
%C Conjecture tested hold up to n=2355. Further test is still running
%C The Mathematica program gives the first 100 terms.
%C Terms for all n are tend to be small integers.
%C 4^n+/-4^k+/-1=2^2n+/-2^2k+/-1
%e n=1, 2=4^1-4^0-4^0, 1 prime found, so a(1)=1;
%e n=2, 11=4^2-4^1-1; 13=4^2-4^1+1; 19=4^2+4^1-1, 3 primes found, so a(2)=3;
%e ...
%e n=13, 67043329=4^13-4^8+1; 67104769=4^13-4^6+1; 67108859=4^13-4^1-1; 67108879=4^13+4^2-1, 4 primes found, so a(13)=4;
%t b = 4; Table[c1 = b^i; cs = {};
%t Do[c2 = b^j; cp = c1 + c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
%t cp = c1 + c2 - 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
%t cp = c1 - c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
%t cp = c1 - c2 - 1;
%t If[PrimeQ[cp], cs = Union[cs, {cp}]], {j, 0, i - 1}];
%t ct = Length[cs]; ct, {i, 1, 100}]
%Y Cf. A196697, A196698.
%K nonn,easy
%O 1,2
%A _Lei Zhou_, Oct 06 2011
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