OFFSET
1,1
COMMENTS
Original name: Starting with the fraction 1/1, this sequence gives the prime numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 10 times bottom to get the new top.
Conjecture: Starting with 1/1, there are infinitely many primes in the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 2k times bottom to get the new top, for k>=1.
a(12) has 5304 digits and is not included here. - Bill McEachen, Jan 22 2023
a(12) = A002535(8563) = 1.0733...*10^5303. - Amiram Eldar, Jun 30 2024
REFERENCES
John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.
FORMULA
Given t(0)=1, b(0)=1 then for i = 1, 2, ..., t(i)/b(i) = (t(i-1) + 10*b(i-1)) /(t(i-1) + b(i-1)), and sequence consists of the t(i) that are prime.
EXAMPLE
MATHEMATICA
Select[Numerator/@NestList[(10Denominator[#]+Numerator[#])/ (Denominator[#]+ Numerator[#])&, 1/1, 200], PrimeQ] (* Harvey P. Dale, Sep 15 2011 *)
Select[LinearRecurrence[{2, 9}, {1, 1}, 150], PrimeQ] (* Amiram Eldar, Jun 30 2024 *)
PROG
(PARI) \\ k=mult, typ=1 num, 2 denom. output prime num or denom
primenum(n, k, typ) = {local(a, b, x, tmp, v); a=1; b=1; for(x=1, n, tmp=b; b=a+b; a=k*tmp+a; if(typ==1, v=a, v=b); if(isprime(v), print1(v, ", "); ) ); print(); print(a/b+.)}
primenum(100, 10, 1)
(Python)
from sympy import isprime
from itertools import islice
from fractions import Fraction
def agen(): # generator of terms
f = Fraction(1, 1)
while True:
n, d = f.numerator + 10*f.denominator, f.numerator + f.denominator
if isprime(n): yield n
f = Fraction(n, d)
print(list(islice(agen(), 11))) # Michael S. Branicky, Jan 23 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, Oct 02 2005
EXTENSIONS
a(11) from Michel Marcus, Jan 23 2023
Name simplified by Sean A. Irvine, Feb 25 2023
STATUS
approved