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A057732 Numbers n such that 2^n + 3 is prime. 27
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 (list; graph; refs; listen; history; text; internal format)
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 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

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=(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

CROSSREFS

Cf. A050414 (2^n-3 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

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Last modified December 8 04:27 EST 2016. Contains 278902 sequences.