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

1,1

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

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Cf. A046328, A046408.

Subsequence of A188650.

Sequence in context: A084994 A046328 A196104 * A229129 A272436 A338309

Adjacent sequences: A046373 A046374 A046375 * A046377 A046378 A046379

Keyword

nonn,base

Author

Patrick De Geest, Jun 15 1998

Extensions

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

More terms from Lars Blomberg, Nov 06 2015

Status

approved