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%I #46 Jun 30 2024 03:22:22
%S 11,31,601,10711,45281,3245551,4057691201,87818089575031,
%T 813086055916584907683448771376472778745411281,
%U 16071419731004292876206308878779566599797733387541964081866111137961,2259503969983505641049567911781316556859822340375755577282633545849516496717511
%N Primes in A002535.
%C 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.
%C 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.
%C a(12) has 5304 digits and is not included here. - _Bill McEachen_, Jan 22 2023
%C a(12) = A002535(8563) = 1.0733...*10^5303. - _Amiram Eldar_, Jun 30 2024
%D John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.
%F 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.
%e 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.
%t Select[Numerator/@NestList[(10Denominator[#]+Numerator[#])/ (Denominator[#]+ Numerator[#])&,1/1,200],PrimeQ] (* _Harvey P. Dale_, Sep 15 2011 *)
%t Select[LinearRecurrence[{2, 9}, {1, 1}, 150], PrimeQ] (* _Amiram Eldar_, Jun 30 2024 *)
%o (PARI) \\ k=mult,typ=1 num,2 denom. output prime num or denom
%o 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+.)}
%o primenum(100, 10, 1)
%o (Python)
%o from sympy import isprime
%o from itertools import islice
%o from fractions import Fraction
%o def agen(): # generator of terms
%o f = Fraction(1, 1)
%o while True:
%o n, d = f.numerator + 10*f.denominator, f.numerator + f.denominator
%o if isprime(n): yield n
%o f = Fraction(n, d)
%o print(list(islice(agen(), 11))) # _Michael S. Branicky_, Jan 23 2023
%Y Cf. A002534, A002535.
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, Oct 02 2005
%E a(11) from _Michel Marcus_, Jan 23 2023
%E Name simplified by _Sean A. Irvine_, Feb 25 2023