A244423 Nonprime palindromes n such that the product of divisors of n is also a palindrome.

1, 4, 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

Derek Orr, Jun 27 2014

Keywords

• nonn
• base

Comments

Primes trivially satisfy this property and are therefore not included in the sequence.

These are the palindromes in A244411.

Examples

			The divisors of 22 are 1, 2, 11 and 22. 1*2*11*22 = 484 is a palindrome. Since 22 is also a palindrome, 22 is a member of this sequence.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..49
• Eric Weisstein's World of Mathematics, Palindromic Number
• Index entries for sequences related to palindromes

Crossrefs

Cf. A244411, A002113.

Programs

Mathematica

palQ[n_] := Block[{d = IntegerDigits@ n}, Reverse@ d == d]; lim = 15000000; Select[Complement[Range@ lim, Prime@ Range@ PrimePi@ lim], And[palQ@ #, palQ[Times @@ Divisors@ #]] &] (* Michael De Vlieger, Aug 25 2015 *)
Select[Range[200002*10^5],!PrimeQ[#]&&AllTrue[{#,Times@@Divisors[#]},PalindromeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 28 2020 *)

PARI

rev(n)={r="";dig=digits(n);for(i=1,#dig,r=concat(Str(dig[i]),r));return(eval(r))}
for(n=1,10^8,if(rev(n)==n&&(!isprime(n)), d=divisors(n);ss=prod(j=1,#d,d[j]);if(ss==rev(ss),print1(n,", "))))

PARI

/* david(n) returns the n-th palindrome from David A. Corneth, Jun 06 2014 */
david(n)={my(d, i, r); r=vector(#digits(n-10^(#digits(n\11)))+#digits(n\11)); n=n-10^(#digits(n\11)); d=digits(n); for(i=1, #d, r[i]=d[i]; r[#r+1-i]=d[i]); sum(i=1, #r, 10^(#r-i)*r[i])}
rev(n)={r="";dig=digits(n);for(i=1,#dig,r=concat(Str(dig[i]),r));return(eval(r))}
for(n=2,10^6,pal=david(n);if(!isprime(pal),d=divisors(pal);ss=prod(j=1,#d,d[j]);if(ss==rev(ss),print1(pal,", "))))

Python

import sympy
from sympy import isprime
from sympy import divisors
def rev(n):
  r = ""
  for i in str(n):
    r = i + r
  return int(r)
def a():
  for n in range(1,10**8):
    if rev(n) == n and not isprime(n):
      p = 1
      for i in divisors(n):
        p*=i
      if rev(p)==p:
        print(n,end=', ')
a()

Python

from sympy import divisor_count, sqrt
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 = 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 = divmod(k,b)
                    m = b*m + r
                yield y*n2 + b*m + k
A244423_list = [1]
for n in palgen(6):
    d = divisor_count(n)
    if d > 2:
        q, r = divmod(d,2)
        s = str(n**q*(sqrt(n) if r else 1))
        if s == s[::-1]:
            A244423_list.append(n) # Chai Wah Wu, Aug 25 2015

Extensions

Edited name by Chai Wah Wu, Aug 25 2015