A196778 a(n) is the number of primes in the form of 4^n+/-4^k+/-1, while 0 <= k < n.

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, 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, 10, 11, 13, 12, 6, 12, 9, 4, 10, 12, 13, 8, 14, 7, 2, 13, 7

Offset: 1

Author

Lei Zhou, Oct 06 2011

Keywords

• nonn
• easy

Comments

Conjecture: all elements of this sequence is greater than 0.

Conjecture tested hold up to n=2355. Further test is still running

The Mathematica program gives the first 100 terms.

Terms for all n are tend to be small integers.

4^n+/-4^k+/-1=2^2n+/-2^2k+/-1

Examples

			n=1, 2=4^1-4^0-4^0, 1 prime found, so a(1)=1;
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;
...
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;
		

Crossrefs

Cf. A196697, A196698.

Programs

Mathematica

b = 4; Table[c1 = b^i; cs = {};
Do[c2 = b^j; cp = c1 + c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 + c2 - 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 - c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 - c2 - 1;
  If[PrimeQ[cp], cs = Union[cs, {cp}]], {j, 0, i - 1}];
ct = Length[cs]; ct, {i, 1, 100}]