A111015 Primes in A002535.

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

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

References

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

Links

Table of n, a(n) for n=1..11.

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

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.

Mathematica

Select[Numerator/@NestList[(10Denominator[#]+Numerator[#])/ (Denominator[#]+ Numerator[#])&, 1/1, 200], PrimeQ] (* Harvey P. Dale, Sep 15 2011 *)

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

Cf. A002534, A002535.

Sequence in context: A328715 A163763 A201808 * A143765 A023306 A068839

Adjacent sequences: A111012 A111013 A111014 * A111016 A111017 A111018

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