A292890 Primes of the form 2^r * 17^s - 1.

3, 7, 31, 67, 127, 271, 577, 1087, 2311, 8191, 78607, 131071, 524287, 1114111, 2367487, 2672671, 17825791, 42762751, 90870847, 606076927, 2147483647, 5151653887, 5815734271, 9697230847, 329705848831, 474351505987, 700624928767, 892896952447, 1168231104511, 2482491097087

Offset: 1

Author

Muniru A Asiru, Sep 26 2017

Keywords

• nonn

Comments

Primes of the forms 2^r * b^s - 1 where b = 1, 5, 7, 11, 13 are A000668 (Mersenne prime exponents), A077313, A077314, A077315 and A173062. When b = 3 we get A005105 with initial term 2.

For n > 1, all terms are congruent to 1 (mod 3).

Also, these are prime numbers p for which (34^p)/(p+1) is an integer.

Examples

			With n = 1, a(1) = 2^2 * 17^0 - 1 = 3.
With n = 4, a(4) = 2^2 * 17^1 - 1 = 67.
list of (r, s): (2, 0), (3, 0), (5, 0), (2, 1), (3, 1), (7, 0), (4, 1), (1, 2), (6, 1), (3, 2), (13, 0), (4, 3), (17, 0), (19, 0), (16, 1), (13, 2), (5, 4), (20, 1), (9, 4), (6, 5).
		

Crossrefs

Cf. Sequences of primes of the forms 2^n * q^s - 1: A000668 (q = 1), A005105 (q = 3), A077313 (q = 5), A077314 (q = 7), A077315 (q = 11), A173062 (q = 13).

Programs

GAP

K:=10^7+1;; # to get all terms <= K.
A:=Filtered(Filtered([1..K], i->i mod 3=1),IsPrime);;    I:=[17];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A292890:=Concatenation([3],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]));

PARI

isok(p) = isprime(p) && (denominator((34^p)/(p+1)) == 1); \\ Michel Marcus, Sep 27 2017

Extensions

More terms from Jinyuan Wang, Feb 23 2020