A291438 Smallest prime of the form (2*n)*3^m + 1 for some m >= 0, or -1 if no such prime exists.

3, 5, 7, 73, 11, 13, 43, 17, 19, 61, 23, 73, 79, 29, 31, 97, 103, 37, 3079, 41, 43, 397, 47, 433, 151, 53, 163, 1102249, 59, 61, 5023, 193, 67, 613, 71, 73, 223, 229, 79, 241, 83, 757, 6967, 89, 271, 277, 283, 97, 883, 101, 103, 313, 107, 109, 331, 113, 3079

Offset: 1

Author

Martin Renner, Aug 23 2017

Keywords

• nonn

Comments

There exist even integers 2*n such that (2*n)*3^m + 1 is always composite.

It is conjectured that the smallest one is 125050976086 = A123159(3), therefore a(62525488043) = -1.

For the corresponding numbers m see A291437.

Examples

			a(4) = 73 because 8*3^2 + 1 = 73 is the smallest prime of this form, since 8*3^0 + 1 = 9 and 8*3^1 + 1 = 25 are not prime.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A057025, A123159, A291437.

Programs

Maple

a:=[]:
for n from 1 to 10^3 do
  t:=-1:
  for m from 0 to 10^3 do # this max value of m is sufficient up to n=10^3
    if isprime((2*n)*3^m+1) then t:=m: break: fi:
  od:
  a:=[op(a),(2*n)*3^t+1]:
od:
a;

Mathematica

Table[If[# < 0, #, 1 + 2 n*3^#] &@ SelectFirst[Range[0, 10^3], PrimeQ[2 n*3^# + 1] &] /. k_ /; MissingQ@ k -> -1, {n, 60}] (* Michael De Vlieger, Aug 23 2017 *)

PARI

a(n) = {my(m = 0); while (!isprime(p=(2*n)*3^m + 1), m++); p;} \\ Michel Marcus, Aug 25 2017