A370447 Palindromic prime numbers that consist only of the digits {0,1,6,8,9} and which remain palindromic primes when their digits are rotated by 180 degrees.

11, 101, 181, 16661, 18181, 19991, 1008001, 1160611, 1180811, 1190911, 1688861, 1880881, 1881881, 1988891, 100111001, 100888001, 101616101, 101919101, 106111601, 106191601, 108101801, 109111901, 109161901, 110111011, 111010111, 111181111, 116010611, 116696611

Offset: 1

Author

Jean-Marc Rebert, Feb 18 2024

Keywords

• nonn
• base

Comments

10886968801 is the least palindromic prime of this sequence for which the set of digits is {0,1,6,8,9}.

Terms must start and end with digit 1 and be of odd length for n > 1. - Michael S. Branicky, Feb 19 2024

Examples

			16661 becomes 19991 under such a rotation, and both are palindromic primes.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..16934 (all terms < 10^20)
• Carlos Rivera, Puzzle 1163. Palindromic primes that are semiprimes when turned upside down, The Prime Puzzles & Problems Connection.

Crossrefs

Subsequence of palindromes in A007597.

Cf. A002385, A178316.

Programs

PARI

rot(u)=my(v=[]);for(i=1,#u,my(x=u[i]);if(x==6,v=concat(9,v),x==9,v=concat(6,v),vecsearch([0,1,8],x)>0,v=concat(x,v)));v
is(x)=my(u=digits(x),su=Set(u));if(setintersect(su,Set([0,1,6,8,9]))!=su||!isprime(x)||Vecrev(u)!=u,return(0));my(y=fromdigits(rot(u)));return(isprime(y))

Python

from sympy import isprime
from itertools import product, count, islice
def flip180(s): return s[::-1].translate({54:57, 57:54})
def agen(): # generator of terms
    yield 11
    for digits in count(3, 2):
        for rest in product("01689", repeat=digits//2-1):
            for mid in "01689":
                s = "".join(("1",)+rest+(mid,)+rest[::-1]+("1",))
                if isprime(t:=int(s)) and isprime(int(flip180(s))):
                    yield t
print(list(islice(agen(), 28))) # Michael S. Branicky, Feb 19 2024