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

%I A291049 #31 Jun 26 2022 03:07:59
%S A291049 2,3,5,17,137,257,65537,157217,295937,557057,1336337,96550277,
%T A291049 1212153857,2281701377,5473632257,395469930497,1401249857537,
%U A291049 2637646790657,4964982194177,28572702478337,1271035441709057,38280596832649217,1872540629620228097,6634884445436379137
%N A291049 Primes of the form 2^r * 17^s + 1.
%C A291049 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.
%C A291049 Fermat prime exponents r are 0, 1, 2, 4, 8, 16.
%C A291049 For n > 2, all terms are congruent to 5 (mod 6).
%C A291049 Also, these are prime numbers p for which (p*34^p)/(p-1) is an integer.
%H A291049 Robert Israel, <a href="/A291049/b291049.txt">Table of n, a(n) for n = 1..625</a>
%e A291049 With n = 1, a(1) = 2^0 * 17^0 + 1 = 2.
%e A291049 With n = 5, a(5) = 2^3 * 17^1 + 1 = 137.
%e A291049 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).
%p A291049 N:= 10^20: # to get all terms <= N+1
%p A291049 S:= NULL:
%p A291049 for r from 0 to ilog2(N) do
%p A291049   for s from 0 to floor(log[17](N/2^r)) do
%p A291049     p:= 2^r*17^s +1;
%p A291049     if isprime(p) then
%p A291049      S:= S, p
%p A291049     fi
%p A291049 od od:
%p A291049 sort([S]); # _Robert Israel_, Sep 26 2017
%t A291049 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 *)
%o A291049 (GAP)
%o A291049 K:=26*10^7+1;; # to get all terms <= K.
%o A291049 A:=Filtered(Filtered([1,3..K],i-> i mod 6=5),IsPrime);;    I:=[17];;
%o A291049 B:=List(A,i->Elements(Factors(i-1)));;
%o A291049 C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
%o A291049 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]));
%o A291049 (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
%Y A291049 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).
%K A291049 nonn
%O A291049 1,1
%A A291049 _Muniru A Asiru_, Sep 15 2017