A077390 Primes which leave primes at every step if most significant digit and least significant digit are deleted until a one digit or two digit prime is obtained.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 127, 131, 137, 139, 151, 157, 173, 179, 223, 227, 229, 233, 239, 251, 257, 271, 277, 331, 337, 353, 359, 373, 379, 421, 431, 433, 439, 457, 479, 521, 523, 557, 571, 577, 631, 653, 659

Offset: 1

Author

Amarnath Murthy, Nov 07 2002

Keywords

• base
• fini
• nonn

Comments

There are exactly 920720315 such primes, the largest being 9161759674286961988443272139114537477768682563429152377117139 1111313737919133977331737137933773713713973. - Karl W. Heuer, Apr 19 2011

There are exactly 331780864 odd length primes and 588939451 even length primes, the largest odd length prime being

7228828176786792552781668926755667258635743361825711373791931117197999133917737137399993737111177. - Seth A. Troisi, May 07 2019

Zeros are not permitted, otherwise this sequence would potentially be infinite (cf. A077391). - Sean A. Irvine, May 19 2025

Examples

			21313 is a member as 21313, 131 and 3 all are primes.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..12975
• Carlos Rivera, Problem 950: Bi-truncatable primes, The Prime Puzzles & Problems Connection.
• Seth A. Troisi, Selected terms: the first 200 primes, the last 300 primes, and the smallest and largest primes of each length

Crossrefs

Cf. A024770 (right-truncatable primes), A024785 (left-truncatable primes), A137812 (left-or-right truncatable primes).

Cf. A077391.

Programs

Mathematica

msd={1,2,3,4,5,6,7,8,9}; lsd={1,3,7,9}; Clear[p]; p[1]={2,3,5,7}; p[2]={11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97}; p[digits_] := p[digits] = Select[Flatten[Outer[Plus, 10^(digits-1)*msd, 10*p[digits-2], lsd]], PrimeQ]; t={}; k=0; While[Length[t] < 100, k++; t=Join[t, p[k]]]; t (* T. D. Noe, Apr 19 2011 *)
paesQ[n_]:=AllTrue[NestWhileList[FromDigits[Most[Rest[ IntegerDigits[ #]]]]&, n,#>99&],PrimeQ]; Select[Prime[Range[150]],paesQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 01 2015 *)

Python

from itertools import count, islice
from sympy import isprime, primerange
def agen(): # generator of terms
    odds, evens, digits = [2, 3, 5, 7], list(primerange(10, 100)), 3
    yield from odds + evens
    while len(odds) > 0 or len(evens) > 0:
        new = []
        old = odds if digits%2 == 1 else evens
        for first in "123456789":
            for p in old:
                mid = str(p)
                for last in "1379":
                    t = int(first + mid + last)
                    if isprime(t):
                        yield t
                        new.append(t)
        old = new
        if digits%2: odds = old
        else: evens = old
        print("...", digits, time()-time0)
        digits += 1
print(list(islice(agen(), 80))) # Michael S. Branicky, May 06 2022

Extensions

Corrected and extended by T. D. Noe, Apr 19 2011