OFFSET
0,1
COMMENTS
First few pairs (n,k) such that k > n are (1,2), (22,26), (24,39), (65,67), (110,150), (112,140), (135,150), (137,169), ... Also, for n=398 there is an interesting anomaly since k=893 which is > 2n.
Conjecture: for every n there exists a number k < 3n such that k*2^n - 1 is prime. Comment from T. D. Noe: this fails at n=624, where a(n)=2163.
Define sumk = Sum_{n=1..N} k(n), and define sumn = Sum_{n=1..N} n, then as N increases the ratio sumk/sumn tends to log(2)/2 = 0.3465735.... so on average k(n) is about 0.35*n and seems to be always < 3.82*n or 11*log(2)/2. - Pierre CAMI, Feb 27 2009
a(n) = 1 if and only if n is in A000043. - Felix Fröhlich, Sep 14 2014
LINKS
Pierre CAMI, Table of n, a(n) for n = 0..3000
FORMULA
a(n) << 19^n by Xylouris's improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013
MATHEMATICA
k2np[n_]:=Module[{k=1, x=2^n}, While[!PrimeQ[k x-1], k++]; k]; Array[ k2np, 90, 0] (* Harvey P. Dale, Nov 19 2011 *)
PROG
(PARI) lim=10^9; for(n=0, 200, k=1; i=0; while(k < lim, if(ispseudoprime(k*2^n-1), print1(k, ", "); i++; break({1})); if(i==0 && k >= lim-1, print1(">", lim, ", "); i=0); k++)) \\ Felix Fröhlich, Sep 20 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Jason Earls, Aug 13 2003
STATUS
approved