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

1,1

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.

Links

Table of n, a(n) for n=1..31.

Example

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.

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 *)

Prog

(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

Crossrefs

Subsequence of A006567 (emirps).

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

Sequence in context: A205822 A251136 A213318 * A097648 A188663 A250711

Adjacent sequences: A346023 A346024 A346025 * A346027 A346028 A346029

Keyword

nonn,base

Author

Lars Blomberg, Jul 14 2021

Status

approved