A056815 Primes with prime "look and say" descriptions.

3, 7, 17, 23, 113, 127, 137, 193, 199, 223, 233, 271, 311, 313, 331, 359, 367, 373, 431, 433, 439, 463, 479, 499, 503, 523, 587, 607, 641, 677, 691, 733, 757, 773, 797, 809, 821, 823, 829, 853, 919, 997, 1009, 1069, 1123, 1129, 1171, 1181, 1187, 1223, 1277

Offset: 1

Author

Joseph L. Pe, Jan 30 2003

Keywords

• base
• nonn

Comments

The "look and say" descriptions of some of these primes are themselves also terms of this sequence (for example, the one for 373). - Alonso del Arte, Mar 01 2012

Examples

			193 is prime and its "look and say" description A045918(193) = 111913, is also prime, so 193 belongs to the sequence.
		

References

• David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005): 41.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A045918, A005150, A127175, A127176.

Cf. A127355.

Programs

Haskell

a056815 n = a056815_list !! (n-1)
a056815_list = filter ((== 1) . a010051' . a045918) a000040_list
-- Reinhard Zumkeller, Apr 14 2014

Mathematica

LookAndSayA[ n_] := FromDigits@Flatten@((Through[ {Length, First}[ # ] ] &) /@ Split@IntegerDigits@n); Select[Prime@Range[210], PrimeQ@LookAndSayA@# &] (* Ray Chandler, Jan 12 2007 *)
(* Emmanuel Vantieghem, Jan 26 2012, reports that the above Mma program is incorrect, because the LookAndSayA function can give wrong answers. Here is a better function (b and c to be substituted by suitable numbers): *)
LookAndSayA[n_] := FromDigits@Flatten@(IntegerDigits/@Flatten@
((Through[{Length, First}[#]]&)/@Split@IntegerDigits@n)); W=Select[Prime@Range[b, c], PrimeQ@LookAndSayA@#&]
(* Robert G. Wilson v then commented (Jan 27 2012) that the following version is cleaner: *)
LookAndSayA[n_] := FromDigits@ Flatten@ IntegerDigits@ Flatten[
Through[{Length, First}[#]] & /@ Split@ IntegerDigits@ n]

PARI

forprime(p=1,999, isprime(A045918(p)) & print1(p","))  \\ M. F. Hasler, Jan 27 2012

Python

from sympy import isprime, sieve
from itertools import groupby, islice
def LS(n): return int(''.join(str(len(list(g)))+k for k, g in groupby(str(n))))
def agen(): yield from (p for p in sieve if isprime(LS(p)))
print(list(islice(agen(), 51))) # Michael S. Branicky, Feb 03 2023

Formula

A010051(a(n)) * A010051(A045918(a(n))) = 1. - Reinhard Zumkeller, Apr 14 2014

Extensions

Discussion of Mma program added Feb 02 2012 by N. J. A. Sloane