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A035050
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a(n) is the smallest k such that k*2^n + 1 is prime.
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21
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1, 1, 1, 2, 1, 3, 3, 2, 1, 15, 12, 6, 3, 5, 4, 2, 1, 6, 3, 11, 7, 11, 25, 20, 10, 5, 7, 15, 12, 6, 3, 35, 18, 9, 12, 6, 3, 15, 10, 5, 6, 3, 9, 9, 15, 35, 19, 27, 15, 14, 7, 14, 7, 20, 10, 5, 27, 29, 54, 27, 31, 36, 18, 9, 12, 6, 3, 9, 31, 23, 39, 39, 40, 20, 10, 5, 58, 29, 15, 36, 18, 9, 13
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OFFSET
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0,4
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COMMENTS
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From Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jun 05 2010: (Start)
If a(i) = 2 * m then a(i+1) = m.
Proof: (I) a(i) = 2*m, 2*m * 2^i + 1 = m*2^(i+1) + 1 prime, so a(i+1) <= m;
(II) if a(i+1) = m-d for an integer d > 0, (m-d) * 2^(i+1) + 1 = (2*m-2*d) * 2^i + 1 prime;
(2m-2d) < 2m contradiction to a(i) = 2 * m, d = 0.
(End)
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LINKS
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FORMULA
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EXAMPLE
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a(3)=2 because 1*2^3 + 1 = 9 is composite, 2*2^3 + 1 = 17 is prime.
a(99)=219 because 2^99k + 1 is not prime for k=1,2,...,218. The first term which is not a composite number of this arithmetic progression is 2^99*219 + 1.
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MATHEMATICA
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a = {}; Do[k = 0; While[ ! PrimeQ[k 2^n + 1], k++ ]; AppendTo[a, k], {n, 0, 100}]; a (* Artur Jasinski *)
Table[Module[{k=1, n2=2^n}, While[!PrimeQ[k*n2+1], k++]; k], {n, 0, 90}] (* Harvey P. Dale, May 25 2024 *)
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PROG
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(PARI) a(n) = {my(k = 1); while (! isprime(2^n*k+1), k++); k; }
(Magma) sol:=[]; m:=1; for n in [0..82] do k:=0; while not IsPrime(k*2^n+1) do k:=k+1; end while; sol[m]:=k; m:=m+1; end for; sol; // Marius A. Burtea, Jun 05 2019
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CROSSREFS
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Analogous case is A034693. Special subscripts (n's for a(n)=1) are the exponents of known Fermat primes: A000215. See also Fermat numbers A000051.
Cf. A007522, A057778, A080076, A085427, A087522, A126717, A127575, A127576, A127577, A127578, A127580, A127581, A127586.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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