A293074 Primes of the form 2^q * 3^r * 11^s - 1.

2, 3, 5, 7, 11, 17, 23, 31, 43, 47, 53, 71, 107, 127, 131, 191, 197, 241, 263, 383, 431, 593, 647, 863, 967, 971, 1151, 1187, 1451, 1583, 2111, 2591, 2903, 3167, 4373, 4751, 5323, 5807, 6143, 6911, 7127, 8191, 8447, 8747, 10691, 12671, 13121, 15551, 15971, 21383, 23327

Offset: 1

Author

Muniru A Asiru, Oct 01 2017

Keywords

• nonn

Comments

Mersenne primes A000668 occur when (q, r, s) = (q, 0, 0) with q > 0.

a(2) = 3 is a Mersenne prime but a(3) = 5 is not a Mersenne prime.

For n > 2, all terms = {1, 5} mod 6.

Examples

			3 = a(2) = 2^2 * 3^0 * 11^0 - 1.
131 = a(15) = 2^2 * 3^1 * 11^1 - 1.
list of (q, r, s): (0, 1, 0), (2, 0, 0), (1, 1, 0), (3, 0, 0), (2, 1, 0), (1, 2, 0), (3, 1, 0), (5, 0, 0), (2, 0, 1), (4, 1, 0), (1, 3, 0), ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000668, A005105, Primes of the form 2^q * 3^r * b^s - 1: A293194 (b = 5), A293199 (b = 7).

Programs

GAP

K:=10^5+1;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;    I:=[3,11];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A293074:=Concatenation([2],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]));

Maple

N:= 10^5: # to get all terms < N
S:=select(isprime, {seq(seq(seq(2^q*3^r*11^s-1, q=0..ilog2(floor(N/3^r/11^s))),r=0..floor(log[3](N/11^s))),s=0..floor(log[11](N)))}):
sort(convert(S,list)); # Robert Israel, Oct 03 2017

Mathematica

With[{nn=20},Take[Select[Union[Flatten[Table[2^q 3^r 11^s-1,{q,0,nn},{r,0,nn},{s,0,nn}]]],PrimeQ],60]] (* Harvey P. Dale, May 12 2019 *)