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A057732
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Numbers n such that 2^n + 3 is prime.
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32
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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.
A number n is in this sequence iff A062709(n) is in A057733; this is the case iff A257273(n) is in A125246. - M. F. Hasler, Apr 27 2015
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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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Table of n, a(n) for n=1..58.
Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Henri & Renaud Lifchitz, PRP 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^n+3; S0=(N-5)/2; s(0)=S0; s(i+1)=s(i)^2-2 modulo N; if s(n-1) == S0 then N is prime. - Tony Reix, Aug 27 2015
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MATHEMATICA
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Select[Range[10000], PrimeQ[2^# + 3] &] (* Vincenzo Librandi, Apr 27 2015 *)
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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
(MAGMA) [n: n in [0..1000] | IsPrime(2^n+3)]; // Vincenzo Librandi, Apr 27 2015
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CROSSREFS
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Cf. A050414 (2^n-3 is prime).
Sequence in context: A057128 A018534 A018276 * A092591 A287924 A039947
Adjacent sequences: A057729 A057730 A057731 * A057733 A057734 A057735
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KEYWORD
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nice,nonn
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AUTHOR
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G. L. Honaker, Jr., Oct 29 2000
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EXTENSIONS
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More terms from Jason Earls, Jul 18 2001 and Mike Oakes, Jul 28 2001
Terms 47 to 50 from Donovan Johnson 2006, verified by Paul Bourdelais, Mar 22 2012
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
a(52) to a(54) correspond to probable primes discovered by Paul Bourdelais, Jun 18 2019
a(55) corresponds to a probable prime discovered by Paul Bourdelais, Jul 16 2019
a(56) corresponds to a probable prime discovered by Paul Bourdelais, Apr 22 2020
a(57) corresponds to a probable prime discovered by Paul Bourdelais, Jun 12 2020
a(58) corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020
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STATUS
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approved
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