

A057732


Numbers n such that 2^n + 3 is prime.


23



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
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OFFSET

1,2


COMMENTS

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


REFERENCES

Mike Oakes, posting to primenumbers(AT)yahoogroups.com on Jul 08 2001


LINKS

Table of n, a(n) for n=1..51.
Henri & Renaud Lifchitz, PRP Records.


FORMULA

Here is an LLTlike algorithm, using a cycle of the digraph x^22 modulo N, that finds terms of this sequence generating a PRP (PRobable Prime) of A057733 numbers: N=2^n+3; S0=(N5)/2; s(0)=S0; s(i+1)=s(i)^22 modulo N; if s(n1) == S0 then N is prime.  Tony Reix, Aug 27 2015


MATHEMATICA

Clear[f, n]; f[n_]:=PrimeQ[2^n+3]; lst={}; Do[If[f[n], AppendTo[lst, n]], {n, 2000}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 02 2009 *)
Select[Range[10000], PrimeQ[2^# + 3] &] (* Vincenzo Librandi, Apr 27 2015 *)


PROG

(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=(N5)/2 ; x=S ; for(j=1, n1, x=Mod(x^22, 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


CROSSREFS

Cf. A050414 (2^n3 is prime).
Sequence in context: A057128 A018534 A018276 * A092591 A039947 A096477
Adjacent sequences: A057729 A057730 A057731 * A057733 A057734 A057735


KEYWORD

nice,nonn


AUTHOR

G. L. Honaker, Jr., Oct 29 2000


EXTENSIONS

More terms from Jason Earls (zevi_35711(AT)yahoo.com), 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


STATUS

approved



