A261534 Nonprime palindromes n with only the digits 0, 1, 2 such that the product of divisors of n is also a palindrome.

1, 22, 111, 121, 202, 1001, 1111, 10001, 10201, 11111, 100001, 1000001, 1001001, 1012101, 1100011, 1101011, 1111111, 10000001, 100000001, 101000101, 110000011, 200010002, 10000000001, 10011111001, 11000100011, 11001010011, 11100100111, 11101010111, 20000100002

Offset: 1

Author

Chai Wah Wu, Aug 31 2015

Keywords

• nonn
• base

Comments

A subsequence of A244423.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..203

Crossrefs

Cf. A244411, A244423.

Programs

Mathematica

lim = 1000000; palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; c = Complement[Range@ lim, Prime@ Range@ PrimePi@ lim]; t = Select[c, Total@ Take[RotateRight@ DigitCount@ #, -7] == 0 &]; Select[t, palQ[Times @@ Divisors@ #] &] (* Michael De Vlieger, Sep 02 2015 *)
Rest[Select[FromDigits/@Tuples[{0,1,2},11],!PrimeQ[#]&&AllTrue[{#,Times@@ Divisors[ #]},PalindromeQ]&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 02 2020 *)

Python

from _future_ import division
from sympy import divisor_count
from gmpy2 import isqrt, t_divmod, digits
def palgen(l,b=10): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            n = b**(x-1)
            n2 = n*b
            for y in range(n,n2):
                k, m = y//b, 0
                while k >= b:
                    k, r = t_divmod(k,b)
                    m = b*m + r
                yield y*n + b*m + k
            for y in range(n,n2):
                k, m = y, 0
                while k >= b:
                    k, r = t_divmod(k,b)
                    m = b*m + r
                yield y*n2 + b*m + k
A261534_list = [1]
for m in palgen(17,3):
    n = int(digits(m,3))
    d = int(divisor_count(n))
    if d > 2:
        q, r = t_divmod(d,2)
        s = digits(n**q*(isqrt(n) if r else 1))
        if s == s[::-1]:
            A261534_list.append(n)