A173236 Primes of the form 2^r * 13^s + 1.

2, 3, 5, 17, 53, 257, 677, 3329, 13313, 35153, 65537, 2768897, 13631489, 2303721473, 3489660929, 4942652417, 11341398017, 10859007357953, 1594691292233729, 31403151600910337, 310144109150467073, 578220423796228097

Offset

1,1

Comments

Necessarily r is even (elementary proof by induction).

s=0 is (trivial) case of 2 and the known five Fermat primes: 2, 3, 5, 17, 257, 65537 (A092506).

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

References

Emil Artin: Galoissche Theorie, Verlag Harri Deutsch, Zuerich, 1973

Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications, 2005

Paulo Ribenboim, Wilfrid Keller, Joerg Richstein: Die Welt der Primzahlen, Springer-Verlag GmbH Berlin, 2006

Links

Table of n, a(n) for n=1..22.

Example

2^0*13^0 + 1 = 2 = prime(1) => a(1).
2^10*13^1 + 1 = 13313 = prime(1581) => a(9).
list of (r,s): (0,0), (1,0), (2,0), (4,0), (2,1), (8,0), (2,2), (8,1), (10,1), (4,3), (16,0), (14,2), (20,1), (20,3), (28,1), (10,6), (26,2), (10,9), (32,5), (40,4), (10,13), (22,10), (32,8), (48,4), (20,13), (2,18), (28,11), (50,6).

Prog

(GAP)
K:=10^7;; # to get all terms <= K.
A:=Filtered([1..K], IsPrime);;
B:=List(A, i->Factors(i-1));;
C:=[];;  for i in B do if Elements(i)=[2] or Elements(i)=[2, 13] then Add(C, Position(B, i)); fi; od;
A173236:=Concatenation([2], List(C, i->A[i])); # Muniru A Asiru, Sep 10 2017

Crossrefs

Cf. A092506, A005105, A092506, A173062.

Sequence in context: A235636 A268130 A103074 * A268209 A245641 A082979

Adjacent sequences: A173233 A173234 A173235 * A173237 A173238 A173239

Keyword

nonn

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 13 2010

Status

approved