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A196778
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a(n) is the number of primes in the form of 4^n+/-4^k+/-1, while 0 <= k < n.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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;
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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