A014234 Largest prime <= 2^n.

2, 3, 7, 13, 31, 61, 127, 251, 509, 1021, 2039, 4093, 8191, 16381, 32749, 65521, 131071, 262139, 524287, 1048573, 2097143, 4194301, 8388593, 16777213, 33554393, 67108859, 134217689, 268435399, 536870909, 1073741789, 2147483647, 4294967291, 8589934583, 17179869143, 34359738337, 68719476731, 137438953447

Offset: 1

Author

Jud McCranie

Keywords

• nonn

Comments

For n>1 largest prime factor of the denominator of A027611(2^n) = 2^n*(2^n)-th harmonic number. - Alexander Adamchuk, Aug 02 2006

References

• D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 390.

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Fred Curtis, C++ program for A014234
• Harry J. Smith, PrimePi2 - Computes the Prime Pi(x) counting function [Broken link]
• Harry J. Smith, PrimePi2 - Computes the Prime Pi(x) counting function [Cached copy]

Crossrefs

Cf. A000079, A014210.

Cf. A013603 (2^n - a(n)).

See comment for the relationship to A027611.

These primes have indices A007053 = number of primes <= 2^n.

The opposite is A104080, delta A092131, indices A372684.

For squarefree instead of prime we have A372889, indices A143658.

A036378 counts primes between powers of 2, A293697 adds them up.

Cf. A035100, A130739, A211997.

Programs

Maple

a:= n-> prevprime(2^n+1):
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 23 2020

Mathematica

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; Table[ Abs[ PrevPrim[2^n]], {n, 1, 30} ]
Join[{2},NextPrime[2^Range[2,40],-1]] (* Harvey P. Dale, Jun 26 2011 *)

PARI

a(n) = precprime(2^n) \\ Michel Marcus, Aug 08 2013

Extensions

Terms for n=31, n=32 added by Fred Curtis (fred(AT)f2.org), Dec 08 2009