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A270884 Smallest of 4 consecutive prime numbers that when represented as a simple continued fraction, generates prime numbers in the numerator and denominator, when reduced.

%I A270884 #21 Jul 22 2025 04:25:20
%S A270884 41,367,619,659,701,2267,2789,3253,3463,6917,8969,9221,11959,13499,
%T A270884 14431,17359,17851,20143,22283,23669,26107,27847,28547,28879,29537,
%U A270884 32503,32717,32987,37549,40709,40849,41647,45971,47161,49339,51061,51199,52571,53171,53479,58337
%N A270884 Smallest of 4 consecutive prime numbers that when represented as a simple continued fraction, generates prime numbers in the numerator and denominator, when reduced.
%C A270884 Order in which the simple continued fraction generated is important. In this case increasing order.
%H A270884 Amiram Eldar, <a href="/A270884/b270884.txt">Table of n, a(n) for n = 1..10726</a> (terms below 10^8; terms 1..100 from Abhiram R Devesh)
%e A270884 For a = 41, the set is [41, 43, 47, 53] in simple continued fraction is
%e A270884 41 +       1
%e A270884      ----------------
%e A270884        43  +    1
%e A270884             ---------
%e A270884              47 + 1
%e A270884                  ----
%e A270884                   53
%e A270884 When reduced 4398061/107209; where 4398061 and 107209 are both primes.
%t A270884 Select[Prime@ Range[10^4], AllTrue[{Numerator@ #, Denominator@ #} &@ FromContinuedFraction@ Prime@ Range[#, # + 3] &@ PrimePi@ #, PrimeQ] &] (* _Michael De Vlieger_, Apr 02 2016, Version 10 *)
%t A270884 cfpnQ[lst_]:=Module[{fcf=FromContinuedFraction[lst]},AllTrue[{Numerator[ fcf],Denominator[ fcf]},PrimeQ]]; Select[Partition[Prime[ Range[ 5000]],4,1],cfpnQ][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Sep 06 2020 *)
%K A270884 nonn
%O A270884 1,1
%A A270884 _Abhiram R Devesh_, Mar 25 2016