A229971 Palindromes n whose product of proper divisors is a palindrome > 1 and not equal to n.

4, 9, 121, 212, 1001, 10201, 110011, 1100011, 10100101, 11000011, 101000101, 110000011, 1010000101, 1100000011, 10000000001, 10100000101, 1000000000001, 10000000000001, 10011100000111001, 10022212521222001, 10100101110100101, 10101100100110101

Offset

1,1

Comments

Palindromes in the sequence A229970.

Links

Table of n, a(n) for n=1..22.

Formula

A229970 INTERSECT A002113.

Example

The product of the proper divisors of 4 is 2 (also a palindrome, different from 4). So, 4 is a member of this sequence.
The proper divisors of 1001 are 1, 7, 11, 13, 77, 91, and 143. 1*7*11*13*77*91*143 = 1001^3 = 1003003001 (also a palindrome, different from 1001). So, 1001 is a member of this sequence.

Mathematica

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; fQ[n_] := Block[{s = Times @@ Most@ Divisors@ n}, And[palQ@ s, s > 1, s != n]]; Select[Select[Range@ 1000000, palQ], fQ] (* Michael De Vlieger, Apr 06 2015 *)
ppdpQ[n_]:=Module[{pp=Times@@Most[Divisors[n]]}, AllTrue[{n, pp}, PalindromeQ]&&pp>1&&pp!=n]; Select[Range[115*10^4], ppdpQ] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Sep 18 2022 *)

Prog

(Python)
from sympy import divisors
def PD(n):
  p = 1
  for i in divisors(n):
    if i != n:
      p *= i
  return p
def pal(n):
  r = ''
  for i in str(n):
    r = i + r
  return r == str(n)
{print(n, end=', ') for n in range(1, 10**6) if pal(n) and pal(PD(n)) and (PD(n)-1) and PD(n)-n}
# Simplified by Derek Orr, Apr 05 2015
# Syntax error fixed by Robert C. Lyons, Mar 17 2023
(PARI) pal(n)=d=digits(n); Vecrev(d)==d
for(n=1, 10^6, D=divisors(n); p=prod(i=1, #D-1, D[i]); if(pal(n)&&pal(p)&&p-1&&p-n, print1(n, ", "))) \\ Derek Orr, Apr 05 2015

Crossrefs

Cf. A229970, A007956.

Sequence in context: A156317 A115676 A115667 * A158642 A131760 A002779

Adjacent sequences: A229968 A229969 A229970 * A229972 A229973 A229974

Keyword

nonn,base

Author

Derek Orr, Oct 04 2013

Extensions

a(7)-a(22) from Giovanni Resta, Oct 06 2013

Name edited by Derek Orr, Apr 05 2015

Status

approved