A133225 Largest prime <= 2^((n+1)/2).

2, 2, 3, 5, 7, 11, 13, 19, 31, 43, 61, 89, 127, 181, 251, 359, 509, 719, 1021, 1447, 2039, 2887, 4093, 5791, 8191, 11579, 16381, 23167, 32749, 46337, 65521, 92681, 131071, 185363, 262139, 370723, 524287, 741431, 1048573, 1482907, 2097143, 2965819

Offset: 1

Author

Anthony C Robin, Jan 03 2008

Keywords

• nonn

Comments

If one is trying to decide whether an n+1 digit binary number is prime, this is the largest prime for which one needs to test divisibility. For example a six digit number like 110101 must be below 64, so only primes up to 7 are needed to test divisibility. Compare with sequence A132153.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A132153.

Programs

Maple

seq(prevprime(floor(2^((n+1)*1/2))+1),n=1..40); # Emeric Deutsch
A017910 := proc(n) floor(2^(n/2)) ; end: A007917 := proc(n) prevprime(n+1) ; end: A133225 := proc(n) A007917(A017910(n+1)) ; end: seq(A133225(n),n=1..60) ; # R. J. Mathar

Mathematica

PrevPrim[n_] := Block[{k = n}, While[ !PrimeQ@k, k-- ]; k]; f[n_] := PrevPrim@ Floor@ Sqrt[2^(n + 1)]; Array[f, 42] (* Robert G. Wilson v *)
Table[Prime[PrimePi[2^((n + 1)/2)]], {n, 1, 50}] (* Stefan Steinerberger *)
lp[n_]:=Module[{c=2^((n+1)/2)},If[PrimeQ[c],c,NextPrime[c,-1]]]; Array[lp,50] (* Harvey P. Dale, Aug 25 2013 *)

Formula

a(n) = A007917[A017910(n+1)]. - R. J. Mathar

Extensions

More terms from Stefan Steinerberger, R. J. Mathar, Robert G. Wilson v and Emeric Deutsch, Jan 06 2008