A256787 Smallest odd number k such that k*2^(2*n+1)+1 is a prime number.

1, 5, 3, 5, 15, 9, 5, 5, 9, 11, 11, 45, 5, 15, 23, 35, 9, 59, 15, 5, 3, 9, 35, 27, 23, 17, 51, 5, 29, 27, 53, 9, 9, 9, 23, 39, 23, 5, 29, 249, 9, 51, 5, 75, 51, 117, 29, 77, 131, 219, 221, 29, 53, 105, 321, 95, 179, 197, 101, 51, 81, 101, 11, 5, 21, 221, 53

Offset

0,2

Comments

As N increases, (Sum_{n=1..N} a(n))/(Sum_{n=1..N} n) appears to tend to log(2), as can be seen by plotting the first 10000 terms.

This observation is consistent with the prime number theorem as the probability that k*2^n+1 is prime is 1/(n*log(2)+log(k)) so ~ 1/(n*log(2)) as n increases, if k ~ n*log(2) then k/(n*log(2)) ~ 1.

Links

Pierre CAMI, Table of n, a(n) for n = 0..10000

Example

1*2^(2*0+1)+1=3 is prime, so a(0)=1.
1*2^(2*1+1)+1=9 and 3*2^(2*1+1)+1=25 are composite, 5*2^(2*1+1)+1=41 is prime, so a(1)=5.

Maple

for n from 0 to 100 do
R:= 2^(2*n+1);
for k from 1 by 2 do
   if isprime(k*R+1) then A[n]:= k; break fi
od:
od:
seq(A[n], n=0..100); # Robert Israel, Apr 24 2015

Mathematica

f[n_] := Block[{g, i, k}, g[x_, y_] := y*2^(2*x + 1) + 1; Reap@ For[i = 0, i <= n, i++, k = 1; While[Nand[PrimeQ[g[i, k]] == True, OddQ@ k], k++]; Sow@ k] // Flatten // Rest]; f@ 66 (* Michael De Vlieger, Apr 18 2015 *)

Prog

(PFGW & SCRIPT)
SCRIPT
DIM i
DIM n, -1
DIMS t
OPENFILEOUT myf, a(n).txt
LABEL loop1
SET n, n+2
SET i, -1
LABEL loop2
SET i, i+2
SETS t, %d, %d\,; n; i
PRP i*2^n+1, t
IF ISPRP THEN GOTO a
GOTO loop2
LABEL a
WRITE myf, t
GOTO loop1
(PARI) vector(100, n, n--; k=1; while(!isprime(k*2^(2*n+1)+1), k+=2); k) \\ Colin Barker, Apr 10 2015

Crossrefs

Sequence in context: A128008 A265800 A144386 * A265782 A073685 A019212

Adjacent sequences: A256784 A256785 A256786 * A256788 A256789 A256790

Keyword

nonn

Author

Pierre CAMI, Apr 10 2015

Status

approved