A293199 Primes of the form 2^q * 3^r * 7^s - 1.

2, 3, 5, 7, 11, 13, 17, 23, 31, 41, 47, 53, 71, 83, 97, 107, 127, 167, 191, 223, 251, 293, 383, 431, 503, 587, 647, 863, 881, 971, 1151, 1511, 1567, 2267, 2351, 2591, 2687, 3023, 3527, 3583, 4373, 4703, 4801, 6047, 6143

Offset: 1

Author

Muniru A Asiru, Oct 02 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.

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

Examples

			3 is a member because it is a prime number and 2^2 * 3^0 * 7^0 - 1 = 3.
503 is a member because it is a prime number and 2^3 * 3^2 * 7^1 - 1 = 503.
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), ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..4000

Crossrefs

Cf. A000668, A005105, A293194.

Programs

GAP

K:=10^5+1;; # to get all terms <=K
A:=Filtered([1..K],IsPrime);; I:=[3,7];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));
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]));

Maple

N:= 10^4: # for terms <= N
S:= {1}:
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:
sort(convert(select(isprime,map(`-`,S,1)),list)); # Robert Israel, Dec 17 2020