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%I #14 Dec 17 2020 15:17:53
%S 2,3,5,7,11,13,17,23,31,41,47,53,71,83,97,107,127,167,191,223,251,293,
%T 383,431,503,587,647,863,881,971,1151,1511,1567,2267,2351,2591,2687,
%U 3023,3527,3583,4373,4703,4801,6047,6143
%N Primes of the form 2^q * 3^r * 7^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.
%C For n > 2, all terms = {1, 5} mod 6.
%H Robert Israel, <a href="/A293199/b293199.txt">Table of n, a(n) for n = 1..4000</a>
%e 3 is a member because it is a prime number and 2^2 * 3^0 * 7^0 - 1 = 3.
%e 503 is a member because it is a prime number and 2^3 * 3^2 * 7^1 - 1 = 503.
%e list of (q, r, s): (0, 1 ,0), (2, 0, 0), (1, 1, 0), (3, 0, 0), (2, 1, 0), (1, 0, 1), (1, 2, 0), (3, 1, 0),(5, 0, 0), (1, 1, 1), (4, 1, 0), (1, 3, 0), (3, 2, 0), (2, 1, 1), ...
%p N:= 10^4: # for terms <= N
%p S:= {1}:
%p for p in {2,3,7} do S:= map(proc(s) local i; seq(s*p^i,i=0..floor(log[p](N/s))) end proc, S) od:
%p sort(convert(select(isprime,map(`-`,S,1)),list)); # _Robert Israel_, Dec 17 2020
%o (GAP) K:=10^5+1;; # to get all terms <=K
%o A:=Filtered([1..K],IsPrime);; I:=[3,7];;
%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 A293199:=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, A293194.
%K nonn
%O 1,1
%A _Muniru A Asiru_, Oct 02 2017
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