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.

41, 367, 619, 659, 701, 2267, 2789, 3253, 3463, 6917, 8969, 9221, 11959, 13499, 14431, 17359, 17851, 20143, 22283, 23669, 26107, 27847, 28547, 28879, 29537, 32503, 32717, 32987, 37549, 40709, 40849, 41647, 45971, 47161, 49339, 51061, 51199, 52571, 53171, 53479, 58337

Offset: 1

Author

Abhiram R Devesh, Mar 25 2016

Keywords

• nonn

Comments

Order in which the simple continued fraction generated is important. In this case increasing order.

Examples

			For a = 41, the set is [41, 43, 47, 53] in simple continued fraction is
41 +       1
     ----------------
       43  +    1
            ---------
             47 + 1
                 ----
                  53
When reduced 4398061/107209; where 4398061 and 107209 are both primes.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10726 (terms below 10^8; terms 1..100 from Abhiram R Devesh)

Programs

Mathematica

Select[Prime@ Range[10^4], AllTrue[{Numerator@ #, Denominator@ #} &@ FromContinuedFraction@ Prime@ Range[#, # + 3] &@ PrimePi@ #, PrimeQ] &] (* Michael De Vlieger, Apr 02 2016, Version 10 *)
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 *)