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A325148 Squares which can be expressed as the product of a number and its reversal.

%I A325148 #58 Jun 09 2021 23:32:27
%S A325148 0,1,4,9,16,25,36,49,64,81,100,121,400,484,900,1089,1600,1936,2500,
%T A325148 3025,3600,4356,4900,5929,6400,7744,8100,9801,10000,10201,12100,12321,
%U A325148 14641,17161,19881,22801,25921,29241,32761,36481,40000,40804,44944,48400,49284,53824,58564,63504,68644,73984,79524,85264
%N A325148 Squares which can be expressed as the product of a number and its reversal.
%C A325148 The numbers k such that k * rev(k) is a square are in A306273.
%C A325148 The squares of palindromes of A014186 are a subsequence.
%C A325148 The square roots of the first 65 terms of this sequence (from 0 to 160000) are exactly the first 65 terms of A061917. Then a(66) = 162409 = 403^2 and the non-palindrome 403 is the first term of another sequence A325151.
%H A325148 Chai Wah Wu, <a href="/A325148/b325148.txt">Table of n, a(n) for n = 1..10000</a> (terms for n = 1..231 from R. J. Mathar)
%H A325148 Bernard Schott, <a href="/A325148/a325148.pdf">The different ways</a>
%F A325148 Intersection of A305231 and A000290. - _R. J. Mathar_, Aug 09 2019
%e A325148 Zero ways: 169 = 13^2 cannot be equal to k * rev(k).
%e A325148 One way: 400 = 200 * 2; 10201 = 101 * 101; 162409 = 169 * 961.
%e A325148 Two ways: 7683984 = 2772 * 2772 = 1584 * 4851.
%e A325148 Three ways: 6350400 = 14400 * 441 = 25200 * 252 = 44100 * 144.
%p A325148 isA305231 := proc(n)
%p A325148     local d;
%p A325148     for d in numtheory[divisors](n) do
%p A325148         if d = digrev(n/d) then
%p A325148             return true ;
%p A325148         end if;
%p A325148     end do:
%p A325148     false ;
%p A325148 end proc:
%p A325148 n := 1;
%p A325148 for i from 0 to 4000 do
%p A325148     i2 := i^2 ;
%p A325148     if isA305231(i2) then
%p A325148         printf("%d %d\n",n,i2) ;
%p A325148         n := n+1 ;
%p A325148     end if;
%p A325148 end do: # _R. J. Mathar_, Aug 09 2019
%t A325148 {0}~Join~Select[Range[10^3]^2,(d1=Select[Divisors[n=#],#<=Sqrt@n&];Or@@Table[d1[[k]]==(IntegerReverse/@(n/d1))[[k]],{k,Length@d1}])&] (* _Giorgos Kalogeropoulos_, Jun 09 2021 *)
%o A325148 (Python)
%o A325148 from sympy import divisors
%o A325148 A325148_list = [0]
%o A325148 for n in range(10**6):
%o A325148     n2 = n**2
%o A325148     for m in divisors(n2):
%o A325148         if m > n:
%o A325148             break
%o A325148         if m == int(str(n2//m)[::-1]):
%o A325148             A325148_list.append(n2)
%o A325148             break # _Chai Wah Wu_, Jun 09 2021
%Y A325148 Equals A325149 Union A083408.
%Y A325148 Cf. A325149 (only one way), A083408 (at least two ways). A325150 (exactly two ways), A307019 (exactly three ways).
%Y A325148 Subsequences: A014186 (square of palindromes), A076750 (product of a non-palindrome and its reversal, where leading zeros are not allowed).
%Y A325148 Cf. A061917, A325151 (some square roots of this sequence).
%K A325148 nonn,base
%O A325148 1,3
%A A325148 _Bernard Schott_, Apr 03 2019
%E A325148 Definition corrected by _N. J. A. Sloane_, Aug 01 2019