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A057732
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Numbers k such that 2^k + 3 is prime.
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43
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1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 28, 30, 55, 67, 84, 228, 390, 784, 1110, 1704, 2008, 2139, 2191, 2367, 2370, 4002, 4060, 4062, 4552, 5547, 8739, 17187, 17220, 17934, 20724, 22732, 25927, 31854, 33028, 35754, 38244, 39796, 40347, 55456, 58312, 122550, 205962, 235326, 363120, 479844, 685578, 742452, 1213815, 1434400, 1594947, 1875552, 1940812, 2205444
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OFFSET
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1,2
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COMMENTS
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Some of the larger entries may only correspond to probable primes.
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REFERENCES
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Mike Oakes, posting to primenumbers(AT)yahoogroups.com on Jul 08 2001
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LINKS
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Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+3, PRP Top Records.
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FORMULA
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Here is an LLT-like algorithm, using a cycle of the digraph x^2-2 modulo N, that finds terms of this sequence generating a PRP (PRobable Prime) of A057733 numbers: N=2^k+3; S0=(N-5)/2; s(0)=S0; s(i+1)=s(i)^2-2 modulo N; if s(k-1) == S0 then N is prime. - Tony Reix, Aug 27 2015
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EXAMPLE
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For k = 6, 2^6 + 3 = 67 is prime.
For k = 28, 2^28 + 3 = 268435459 is prime.
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MATHEMATICA
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PROG
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(PARI) for(n=1, 2200, if(isprime(2^n+3), print1(n, ", ")));
(PARI) for (n=1, 2, if (isprime(2^n+3), print1(n, ", "))); for(n=3, 100000, N=2^n+3 ; S=(N-5)/2 ; x=S ; for(j=1, n-1, x=Mod(x^2-2, N)) ; if(x==S , print1(n, ", "))) \\ produces terms corresponding to probable primes, see formula; Tony Reix, Aug 27 2015
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CROSSREFS
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Cf. A019434 (primes 2^k+1), this sequence (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196(2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23).
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KEYWORD
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nice,nonn
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AUTHOR
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EXTENSIONS
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a(51) is a probable prime based on trial factoring to 1E9 and PRP testing base 3,5,7 (PFGW v3.3.1). Discovered by Paul Bourdelais, Apr 09 2012
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STATUS
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approved
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