A079637 Smallest prime p with audioactive "echo" of at least n, that is, the finite sequence p_0 = p, p_1 = LookAndSay(p_0), ..., p_n = LookAndSay(p_(n-1)) consists entirely of primes.

2, 3, 7, 233, 233, 233, 19972667609, 75022592087629

Offset: 0

Author

Joseph L. Pe, Jan 30 2003

Keywords

• nonn
• base
• more

Comments

LookAndSay(n) denotes the description of the digits of n. For example, LookAndSay(111223) = 312213. 2. There is no prime < 10^5 with echo = 6.

Examples

			233 is the smallest prime p such that p_0 = 233, p_1 = LookAndSay(233) = 1223, p_2 = LookAndSay(1223) = 112213.
		

Links

• Carlos Rivera, Puzzle 36. Sequences of "descriptive primes", The Prime Puzzles and Problems Connection.
• Carlos Rivera, Puzzle 999. In Memoriam to John Horton Conway, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A045918, A005150.

Programs

Python

from sympy import isprime, nextprime
from itertools import groupby, islice
def LS(n):
    return int(''.join(str(len(list(g)))+k for k, g in groupby(str(n))))
def f(n): return -1 if not isprime(n) else 1 + f(LS(n))
def agen(startn=0, startp=2):
    n, p = startn, startp
    while True:
        fp = f(p)
        while (fp >= n): n += 1; yield p
        p = nextprime(p)
print(list(islice(agen(), 6))) # Michael S. Branicky, Jul 27 2022

Extensions

Corrected by Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 02 2003, who reports there are no more terms < 10^6.

a(6) (found by Walter Schneider) and a(7) from Giovanni Resta, May 09 2020