A058383 Primes of form 1+(2^a)*(3^b), a>0, b>0.

7, 13, 19, 37, 73, 97, 109, 163, 193, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993

Offset: 1

Author

Labos Elemer, Dec 20 2000

Keywords

• nonn

Comments

Prime numbers n such that cos(2*Pi/n) is an algebraic number of a 3-smooth degree, but not a 2-smooth degree. - Artur Jasinski, Dec 13 2006

From Antonio M. Oller-Marcén, Sep 24 2009: (Start)

In this case gcd(a,b) is a power of 2.

A regular polygon of n sides is constructible by paper folding if and only if n=2^r3^sp_1...p_t with p_i being distinct primes of this kind. (End)

Primes in A005109 but not in A092506. - R. J. Mathar, Sep 28 2012

Conjecture: these are the only solutions >=7 to the equation A000010(x) + A000010(x-1) = floor((4*x-3)/3). - Benoit Cloitre, Mar 02 2018

These are also called Pierpont primes. - Harvey P. Dale, Apr 13 2019

Links

• Ray Chandler, Table of n, a(n) for n = 1..8378 (terms < 10^1000, first 1000 terms from T. D. Noe)

Crossrefs

Cf. A033845, A000423, A125866, A217035.

Programs

Maple

N:= 10^10: # to get all terms <= N+1
sort(select(isprime, [seq(seq(1+2^a*3^b, a=1..ilog2(N/3^b)), b=1..floor(log[3](N)))])); # Robert Israel, Mar 02 2018

Mathematica

Do[If[Take[FactorInteger[EulerPhi[2n + 1]][[ -1]],1] == {3} && PrimeQ[2n + 1], Print[2n + 1]], {n, 1, 10000}] (* Artur Jasinski, Dec 13 2006 *)
mx = 1500000; s = Sort@ Flatten@ Table[1 + 2^j*3^k, {j, Log[2, mx]}, {k, Log[3, mx/2^j]}]; Select[s, PrimeQ] (* Robert G. Wilson v, Sep 28 2012 *)
Select[Prime[Range[114000]],FactorInteger[#-1][[All,1]]=={2,3}&] (* Harvey P. Dale, Apr 13 2019 *)

Formula

Primes of the form 1 + A033845(n).