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

%I A229970 #26 Nov 24 2024 20:34:09
%S A229970 4,9,25,49,121,212,1001,2636,10201,17161,22801,32761,36481,97969,
%T A229970 110011,124609,139129,146689,528529,573049,619369,635209,844561,
%U A229970 863041,1100011,10100101,11000011,101000101,106110601,110000011,110271001,112381201,127938721,130210921
%N A229970 Numbers n such that the product of their proper divisors is a palindrome > 1 and not equal to n.
%C A229970 Since the product of proper divisors must be > 1, these terms are necessarily composite. - _Derek Orr_, Apr 05 2015
%H A229970 Charles R Greathouse IV, <a href="/A229970/b229970.txt">Table of n, a(n) for n = 1..62</a>
%e A229970 The product of the proper divisors of 2636 is 6948496 (a palindrome). So, 2636 is a member of this sequence.
%e A229970 The product of the proper divisors of 8 is 8 (a palindrome) but equal to 8. So 8 is not in this sequence.
%p A229970 isA002113 := proc(n)
%p A229970     dgs := convert(n,base,10) ;
%p A229970     for i from 1 to nops(dgs)/2 do
%p A229970         if op(i,dgs) <> op(-i,dgs) then
%p A229970             return false;
%p A229970         end if;
%p A229970     end do:
%p A229970     true ;
%p A229970 end proc:
%p A229970 for n from 4 do
%p A229970     if not isprime(n) then
%p A229970         ppd := A007956(n) ;
%p A229970         if n <> ppd and isA002113(ppd) then
%p A229970             printf("%d,",n);
%p A229970         end if;
%p A229970     end if;
%p A229970 end do: # _R. J. Mathar_, Oct 09 2013
%t A229970 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 *)
%o A229970 (Python)
%o A229970 from sympy import divisors
%o A229970 def PD(n):
%o A229970   p = 1
%o A229970   for i in divisors(n):
%o A229970     if i != n:
%o A229970       p *= i
%o A229970   return p
%o A229970 def pal(n):
%o A229970   r = ''
%o A229970   for i in str(n):
%o A229970     r = i + r
%o A229970   return r == str(n)
%o A229970 {print(n, end=', ') for n in range(1, 10**4) if pal(PD(n)) and (PD(n)-1) and PD(n)-n}
%o A229970 ## Simplified by _Derek Orr_, Apr 05 2015
%o A229970 (PARI) ispal(n)=Vecrev(n=digits(n))==n
%o A229970 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
%o A229970 (PARI) pal(n)=d=digits(n);Vecrev(d)==d
%o A229970 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
%Y A229970 Cf. A007956.
%K A229970 nonn,base
%O A229970 1,1
%A A229970 _Derek Orr_, Oct 04 2013
%E A229970 a(14)-a(18) from _R. J. Mathar_, Oct 09 2013
%E A229970 a(19)-a(34) from _Charles R Greathouse IV_, Oct 09 2013
%E A229970 Definition edited by _Derek Orr_, Apr 05 2015