A346026 Primes that are the first in a run of exactly 6 emirps.

10039, 14891, 39791, 119773, 149561, 162293, 163781, 176903, 181751, 197383, 336689, 392911, 393361, 714361, 715361, 779003, 971141, 995443, 996539, 1165037, 1284487, 1307729, 1447151, 1611877, 1640539, 1789621, 1891147, 3136909, 3150557, 3284447, 3339943

Offset: 1

Author

Lars Blomberg, Jul 14 2021

Keywords

• nonn
• base

Comments

There are large gaps in this sequence because all terms need to begin with 1, 3, 7, or 9 otherwise the reversal is composite.

Examples

			a(1) = 10039 because of the eight consecutive primes 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093 all except 10037 and 10093 are emirps and this is the first such occurrence.
		

Crossrefs

Subsequence of A006567 (emirps).

Cf. A003684, A048052, A048054, A071612, A346022, A346023, A346024, A346025.

Programs

Mathematica

EmQ[n_]:=(s=IntegerReverse@n;PrimeQ@s&&n!=s);
Select[Prime@Range[2,50000],Boole[EmQ/@NextPrime[#,Range[-1,6]]]=={0,1,1,1,1,1,1,0}&] (* Giorgos Kalogeropoulos, Jul 27 2021 *)

Python

from sympy import isprime, nextprime, prime, primerange
def isemirp(p): s = str(p); return s != s[::-1] and isprime(int(s[::-1]))
def aupto(limit, runlength=6):
  alst = []
  pvec = list(primerange(1, prime(runlength+2)+1))
  evec = [int(isemirp(p)) for p in pvec]
  target = [0] + [1 for i in range(runlength)] + [0]
  p = nextprime(pvec[-1])
  while pvec[1] <= limit:
    if evec == target: alst.append(pvec[1])
    pvec = pvec[1:] + [p]; evec = evec[1:] + [isemirp(p)]; p = nextprime(p)
    strp = str(p)
    if strp[0] in "24568": # skip large gaps (p is a prime, not an emirp)
      evec = [0 for i in range(runlength+2)]
      pvec = [0 for i in range(runlength+2)]
      p = nextprime(int(str(int(strp[0])+1)+'0'*(len(strp)-1)))
  return alst
print(aupto(3339943)) # Michael S. Branicky, Jul 14 2021