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, Zürich, 1973.
Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications, 2005.
Paulo Ribenboim, Wilfrid Keller, and Joerg Richstein, Die Welt der Primzahlen, Springer-Verlag GmbH Berlin, 2006.
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
(Python)
from itertools import count, islice
from sympy import isprime, integer_log
def A173236_gen(): # generator of terms
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
kmin = kmax >> 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def g(n):
def f(x): return n+x-sum(((x-1)//13**i).bit_length() for i in range(integer_log(x-1, 13)[0]+1))
return bisection(f, n+1, n+1)
return filter(lambda n:isprime(n), map(g, count(1)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 13 2010
STATUS
approved