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Numbers n such that the product of their proper divisors is a palindrome > 1 and not equal to n.
2

%I #26 Nov 24 2024 20:34:09

%S 4,9,25,49,121,212,1001,2636,10201,17161,22801,32761,36481,97969,

%T 110011,124609,139129,146689,528529,573049,619369,635209,844561,

%U 863041,1100011,10100101,11000011,101000101,106110601,110000011,110271001,112381201,127938721,130210921

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

%C Since the product of proper divisors must be > 1, these terms are necessarily composite. - _Derek Orr_, Apr 05 2015

%H Charles R Greathouse IV, <a href="/A229970/b229970.txt">Table of n, a(n) for n = 1..62</a>

%e The product of the proper divisors of 2636 is 6948496 (a palindrome). So, 2636 is a member of this sequence.

%e The product of the proper divisors of 8 is 8 (a palindrome) but equal to 8. So 8 is not in this sequence.

%p isA002113 := proc(n)

%p dgs := convert(n,base,10) ;

%p for i from 1 to nops(dgs)/2 do

%p if op(i,dgs) <> op(-i,dgs) then

%p return false;

%p end if;

%p end do:

%p true ;

%p end proc:

%p for n from 4 do

%p if not isprime(n) then

%p ppd := A007956(n) ;

%p if n <> ppd and isA002113(ppd) then

%p printf("%d,",n);

%p end if;

%p end if;

%p end do: # _R. J. Mathar_, Oct 09 2013

%t 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 (Python)

%o from sympy import divisors

%o def PD(n):

%o p = 1

%o for i in divisors(n):

%o if i != n:

%o p *= i

%o return p

%o def pal(n):

%o r = ''

%o for i in str(n):

%o r = i + r

%o return r == str(n)

%o {print(n, end=', ') for n in range(1, 10**4) if pal(PD(n)) and (PD(n)-1) and PD(n)-n}

%o ## Simplified by _Derek Orr_, Apr 05 2015

%o (PARI) ispal(n)=Vecrev(n=digits(n))==n

%o 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 (PARI) pal(n)=d=digits(n);Vecrev(d)==d

%o 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 Cf. A007956.

%K nonn,base

%O 1,1

%A _Derek Orr_, Oct 04 2013

%E a(14)-a(18) from _R. J. Mathar_, Oct 09 2013

%E a(19)-a(34) from _Charles R Greathouse IV_, Oct 09 2013

%E Definition edited by _Derek Orr_, Apr 05 2015