OFFSET
1,2
COMMENTS
Conjecture: a(n) exists for every n.
The conjecture is a corollary of Dirichlet's theorem on primes in arithmetic progressions. - Robert Israel, Jan 05 2016
As N increases sum {k, n=1 to N} / sum {n, n=1 to N} tends to 0.818.
If k=1 then n*2^n+1 is a Cullen prime.
Generalized Cullen primes have the form n*b^n+1, I propose to name the primes k*n*2^n-1 generalized Cullen primes of the second type.
LINKS
Pierre CAMI, Table of n, a(n) for n = 1..10000
EXAMPLE
1*1*2^1+1=3 prime so a(1)=1.
1*2*2^2+1=9 composite, 3*2*2^2+1=25 composite, 5*2*2^2+1=41 prime so a(2)=5.
1*3*2^3+1=25 composite, 3*3*2^3=73 prime so a(3)=3.
MAPLE
Q:= proc(m) local k;
for k from 1 by 2 do if isprime(k*m+1) then return k fi od
end proc: seq(Q(n*2^n), n=1..100); # Robert Israel, Jan 05 2016
MATHEMATICA
Table[k = 1; While[!PrimeQ[k*n*2^n + 1], k += 2]; k, {n, 73}] (* Michael De Vlieger, Apr 21 2015 *)
PROG
(PFGW & SCRIPT)
SCRIPT
DIM n, 0
DIM k
DIMS t
OPENFILEOUT myf, a(n).txt
LABEL loop1
SET n, n+1
IF n>3000 THEN END
SET k, -1
LABEL loop2
SET k, k+2
SETS t, %d, %d\,; n; k
PRP k*n*2^n+1, t
IF ISPRP THEN GOTO a
GOTO loop2
LABEL a
WRITE myf, t
GOTO loop1
(PARI) a(n) = k=1; while(!isprime(k*n*2^n+1), k+=2); k \\ Colin Barker, Apr 21 2015
(PFGW) ABC2 $b*$a*2^$a+1 // {number_primes, $b, 1}
a: from 1 to 10000
b: from 1 to 100000 step 2
Charles R Greathouse IV, Apr 24 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Pierre CAMI, Apr 21 2015
STATUS
approved