A111015 Primes in A002535.

11, 31, 601, 10711, 45281, 3245551, 4057691201, 87818089575031, 813086055916584907683448771376472778745411281, 16071419731004292876206308878779566599797733387541964081866111137961, 2259503969983505641049567911781316556859822340375755577282633545849516496717511

Offset: 1

Author

Cino Hilliard, Oct 02 2005

Keywords

• easy
• nonn

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

Examples

			The raw ratios begin 1/1, 11/2, 31/13, 161/44, 601/205, ... = A002535/A002534. Among the numerators, 11, 31, and 601 are primes and are the first three terms here.
		

References

• John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.

Crossrefs

Cf. A002534, A002535.

Programs

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 *)

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

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.

Extensions

a(11) from Michel Marcus, Jan 23 2023

Name simplified by Sean A. Irvine, Feb 25 2023