A291049 Primes of the form 2^r * 17^s + 1.

2, 3, 5, 17, 137, 257, 65537, 157217, 295937, 557057, 1336337, 96550277, 1212153857, 2281701377, 5473632257, 395469930497, 1401249857537, 2637646790657, 4964982194177, 28572702478337, 1271035441709057, 38280596832649217, 1872540629620228097, 6634884445436379137

Offset: 1

Author

Muniru A Asiru, Sep 15 2017

Keywords

• nonn

Comments

Primes of the forms a^r * b^s + 1 where (a, b) = (2,1), (2,3), (2,5), (2,7), (2,11) and (2,13) are A092506, A005109, A077497, A077498, A077499 and A173236.

Fermat prime exponents r are 0, 1, 2, 4, 8, 16.

For n > 2, all terms are congruent to 5 (mod 6).

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

Examples

			With n = 1, a(1) = 2^0 * 17^0 + 1 = 2.
With n = 5, a(5) = 2^3 * 17^1 + 1 = 137.
list of (r,s): (0,0), (1,0), (2,0), (4,0), (3,1), (8,0), (16,0), (5,3), (10,2), (15,1), (4,4), (2,6).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..625

Crossrefs

Cf. Sequences of primes of form 2^n * q^u + 1: A092506 (q=1), A005109 (q=3), A077497 (q=5), A077498 (q=7), A077499 (q=11), A173236 (q=13).

Programs

GAP

K:=26*10^7+1;; # to get all terms <= K.
A:=Filtered(Filtered([1,3..K],i-> i mod 6=5),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)));;
A291049:=Concatenation([2,3],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]));

Maple

N:= 10^20: # to get all terms <= N+1
S:= NULL:
for r from 0 to ilog2(N) do
  for s from 0 to floor(log[17](N/2^r)) do
    p:= 2^r*17^s +1;
    if isprime(p) then
     S:= S, p
    fi
od od:
sort([S]); # Robert Israel, Sep 26 2017

Mathematica

With[{nn = 10^19, q = 17}, Select[Sort@ Flatten@ Table[2^i*q^j + 1, {i, 0, Log[2, nn]}, {j, 0, Log[q, nn/2^i]}], PrimeQ]] (* Michael De Vlieger, Sep 18 2017, after Robert G. Wilson v at A005109 *)

PARI

lista(nn) = my(t, v=List([])); for(r=0, logint(nn, 2), t=2^r; for(s=0, logint(nn\t, 17), if(isprime(t+1), listput(v, t+1)); t*=17)); Vec(vecsort(v)) \\ Jinyuan Wang, Jun 26 2022