A370447 - OEIS

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

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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