A046376 Palindromes with exactly 2 palindromic prime factors (counted with multiplicity), and no other prime factors.

4, 6, 9, 22, 33, 55, 77, 121, 202, 262, 303, 393, 505, 626, 707, 939, 1111, 1441, 1661, 1991, 3443, 3883, 7997, 10201, 13231, 15251, 18281, 19291, 20602, 22622, 22822, 24842, 26662, 28682, 30903, 31613, 33933, 35653, 37673, 38683, 39993, 60206, 60406, 60806

Offset: 1

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn
• base

Comments

Equivalently, semiprime palindromes where both prime factors are palindromes. - Franklin T. Adams-Watters, Apr 11 2011

The sequence "trivially" includes products of palindromic primes p*q where

a) p = 2 or 3 and q has only digits < 4, as q = 11, 101, 131, 10301, 30103, ...

b) p <= 11 and q has only digits 0 and 1, as q = 101 and repunit primes A004022

c) p = 11 and q has only digits spaced out by zeros, as q = 101, 10301, 10501, 10601, 30103, 30203, 30403, 30703, 30803, ... - M. F. Hasler, Jan 04 2022

Examples

			The palindrome 35653 is a term since it has 2 factors, 101 and 353, both palindromic.
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001358 (semiprimes), A002113 (palindromes), A002385 (palindromic primes).

Cf. A046328, A046408.

Subsequence of A188650.

Programs

Mathematica

Take[Select[Times@@@Tuples[Select[Prime[Range[5000]],PalindromeQ],2], PalindromeQ]// Union,50] (* Harvey P. Dale, Aug 25 2019 *)

PARI

{first(N=50, p=1) = vector(N, i, until( bigomega( p=nxt_A002113(p))==2 && vecmin( apply( is_A002113, factor(p)[,1])),); p)} \\ M. F. Hasler, Jan 04 2022

Python

from sympy import factorint
from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal):
    f = factorint(pal)
    return sum(f.values()) == 2 and all(ispal(p) for p in f)
print(list(filter(ok, (p for d in range(1, 6) for p in pals(d) if ok(p))))) # Michael S. Branicky, Aug 14 2022

Formula

Intersection of A002113 and A046368; A188649(a(n)) = a(n). - Reinhard Zumkeller, Apr 11 2011

Extensions

Definition clarified by Franklin T. Adams-Watters, Apr 11 2011

More terms from Lars Blomberg, Nov 06 2015