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

%I A244423 #40 May 22 2025 10:21:39
%S A244423 1,4,22,111,121,202,1001,1111,10001,10201,11111,100001,1000001,
%T A244423 1001001,1012101,1100011,1101011,1111111,10000001,100000001,101000101,
%U A244423 110000011,200010002,10000000001,10011111001,11000100011,11001010011,11100100111,11101010111,20000100002
%N A244423 Nonprime palindromes n such that the product of divisors of n is also a palindrome.
%C A244423 Primes trivially satisfy this property and are therefore not included in the sequence.
%C A244423 These are the palindromes in A244411.
%H A244423 Chai Wah Wu, <a href="/A244423/b244423.txt">Table of n, a(n) for n = 1..49</a>
%H A244423 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PalindromicNumber.html">Palindromic Number</a>
%H A244423 <a href="/index/Pac#palindromes">Index entries for sequences related to palindromes</a>
%e A244423 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.
%t A244423 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 *)
%t A244423 Select[Range[200002*10^5],!PrimeQ[#]&&AllTrue[{#,Times@@Divisors[#]},PalindromeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 28 2020 *)
%o A244423 (PARI) rev(n)={r="";dig=digits(n);for(i=1,#dig,r=concat(Str(dig[i]),r));return(eval(r))}
%o A244423 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,", "))))
%o A244423 (PARI) /* david(n) returns the n-th palindrome from _David A. Corneth_, Jun 06 2014 */
%o A244423 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])}
%o A244423 rev(n)={r="";dig=digits(n);for(i=1,#dig,r=concat(Str(dig[i]),r));return(eval(r))}
%o A244423 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,", "))))
%o A244423 (Python)
%o A244423 import sympy
%o A244423 from sympy import isprime
%o A244423 from sympy import divisors
%o A244423 def rev(n):
%o A244423   r = ""
%o A244423   for i in str(n):
%o A244423     r = i + r
%o A244423   return int(r)
%o A244423 def a():
%o A244423   for n in range(1,10**8):
%o A244423     if rev(n) == n and not isprime(n):
%o A244423       p = 1
%o A244423       for i in divisors(n):
%o A244423         p*=i
%o A244423       if rev(p)==p:
%o A244423         print(n,end=', ')
%o A244423 a()
%o A244423 (Python)
%o A244423 from sympy import divisor_count, sqrt
%o A244423 def palgen(l,b=10): # generator of palindromes in base b of length <= 2*l
%o A244423     if l > 0:
%o A244423         yield 0
%o A244423         for x in range(1,l+1):
%o A244423             n = b**(x-1)
%o A244423             n2 = n*b
%o A244423             for y in range(n,n2):
%o A244423                 k, m = y//b, 0
%o A244423                 while k >= b:
%o A244423                     k, r = divmod(k,b)
%o A244423                     m = b*m + r
%o A244423                 yield y*n + b*m + k
%o A244423             for y in range(n,n2):
%o A244423                 k, m = y, 0
%o A244423                 while k >= b:
%o A244423                     k, r = divmod(k,b)
%o A244423                     m = b*m + r
%o A244423                 yield y*n2 + b*m + k
%o A244423 A244423_list = [1]
%o A244423 for n in palgen(6):
%o A244423     d = divisor_count(n)
%o A244423     if d > 2:
%o A244423         q, r = divmod(d,2)
%o A244423         s = str(n**q*(sqrt(n) if r else 1))
%o A244423         if s == s[::-1]:
%o A244423             A244423_list.append(n) # _Chai Wah Wu_, Aug 25 2015
%Y A244423 Cf. A244411, A002113.
%K A244423 nonn,base
%O A244423 1,2
%A A244423 _Derek Orr_, Jun 27 2014
%E A244423 Edited name by _Chai Wah Wu_, Aug 25 2015