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

1,1

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

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.

Example

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

Prog

(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

Crossrefs

Subsequence of palindromes in A007597.

Cf. A002385, A178316.

Sequence in context: A007597 A061247 A199328 * A068188 A032592 A180404

Adjacent sequences: A370444 A370445 A370446 * A370448 A370449 A370450

Keyword

nonn,base

Author

Jean-Marc Rebert, Feb 18 2024

Status

approved