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%I #21 May 12 2019 13:03:02
%S 2,3,5,7,11,17,23,31,43,47,53,71,107,127,131,191,197,241,263,383,431,
%T 593,647,863,967,971,1151,1187,1451,1583,2111,2591,2903,3167,4373,
%U 4751,5323,5807,6143,6911,7127,8191,8447,8747,10691,12671,13121,15551,15971,21383,23327
%N Primes of the form 2^q * 3^r * 11^s - 1.
%C Mersenne primes A000668 occur when (q, r, s) = (q, 0, 0) with q > 0.
%C a(2) = 3 is a Mersenne prime but a(3) = 5 is not a Mersenne prime.
%C For n > 2, all terms = {1, 5} mod 6.
%H Robert Israel, <a href="/A293074/b293074.txt">Table of n, a(n) for n = 1..10000</a>
%e 3 = a(2) = 2^2 * 3^0 * 11^0 - 1.
%e 131 = a(15) = 2^2 * 3^1 * 11^1 - 1.
%e 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), ...
%p N:= 10^5: # to get all terms < N
%p 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)))}):
%p sort(convert(S,list)); # _Robert Israel_, Oct 03 2017
%t 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 *)
%o (GAP)
%o K:=10^5+1;; # to get all terms <= K.
%o A:=Filtered([1..K],IsPrime);; I:=[3,11];;
%o B:=List(A,i->Elements(Factors(i+1)));;
%o C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
%o 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]));
%Y Cf. A000668, A005105, Primes of the form 2^q * 3^r * b^s - 1: A293194 (b = 5), A293199 (b = 7).
%K nonn
%O 1,1
%A _Muniru A Asiru_, Oct 01 2017
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