A229970 Numbers n such that the product of their proper divisors is a palindrome > 1 and not equal to n.

4, 9, 25, 49, 121, 212, 1001, 2636, 10201, 17161, 22801, 32761, 36481, 97969, 110011, 124609, 139129, 146689, 528529, 573049, 619369, 635209, 844561, 863041, 1100011, 10100101, 11000011, 101000101, 106110601, 110000011, 110271001, 112381201, 127938721, 130210921

Offset: 1

Author

Derek Orr, Oct 04 2013

Keywords

• nonn
• base

Comments

Since the product of proper divisors must be > 1, these terms are necessarily composite. - Derek Orr, Apr 05 2015

Examples

			The product of the proper divisors of 2636 is 6948496 (a palindrome). So, 2636 is a member of this sequence.
The product of the proper divisors of 8 is 8 (a palindrome) but equal to 8. So 8 is not in this sequence.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..62

Crossrefs

Cf. A007956.

Programs

Maple

isA002113 := proc(n)
    dgs := convert(n,base,10) ;
    for i from 1 to nops(dgs)/2 do
        if op(i,dgs) <> op(-i,dgs) then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 4 do
    if not isprime(n) then
        ppd := A007956(n) ;
        if n <> ppd and isA002113(ppd) then
            printf("%d,",n);
        end if;
    end if;
end do: # R. J. Mathar, Oct 09 2013

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[Range@ 1000000, CompositeQ@ # && fQ@ # &] (* Michael De Vlieger, Apr 06 2015 *)

PARI

ispal(n)=Vecrev(n=digits(n))==n
is(n)=my(k=if(issquare(n,&k),k^numdiv(n)/n,n^(numdiv(n)/2-1))); k!=n && k>1 && ispal(k) \\ Charles R Greathouse IV, Oct 09 2013

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(p)&&p-1&&p-n,print1(n,", "))) \\ Derek Orr, Apr 05 2015

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**4) if pal(PD(n)) and (PD(n)-1) and PD(n)-n}
## Simplified by Derek Orr, Apr 05 2015

Extensions

a(14)-a(18) from R. J. Mathar, Oct 09 2013

a(19)-a(34) from Charles R Greathouse IV, Oct 09 2013

Definition edited by Derek Orr, Apr 05 2015