|
| |
|
|
A257378
|
|
Smallest odd number k such that k*n*2^n+1 is a prime number.
|
|
4
|
|
|
|
1, 5, 3, 3, 13, 3, 3, 9, 5, 13, 9, 3, 3, 5, 9, 7, 3, 3, 3, 5, 3, 7, 19, 5, 5, 33, 3, 7, 7, 9, 5, 15, 3, 21, 15, 7, 35, 89, 25, 15, 25, 49, 53, 45, 13, 15, 21, 31, 27, 3, 9, 33, 37, 23, 41, 41, 19, 9, 111, 7, 3, 89, 13, 39, 31, 17, 11, 101, 17, 37, 7, 51, 75
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
|
|
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
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
