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Squares which can be expressed as the product of a number and its reversal.
8

%I #58 Jun 09 2021 23:32:27

%S 0,1,4,9,16,25,36,49,64,81,100,121,400,484,900,1089,1600,1936,2500,

%T 3025,3600,4356,4900,5929,6400,7744,8100,9801,10000,10201,12100,12321,

%U 14641,17161,19881,22801,25921,29241,32761,36481,40000,40804,44944,48400,49284,53824,58564,63504,68644,73984,79524,85264

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

%C The numbers k such that k * rev(k) is a square are in A306273.

%C The squares of palindromes of A014186 are a subsequence.

%C 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 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 Bernard Schott, <a href="/A325148/a325148.pdf">The different ways</a>

%F Intersection of A305231 and A000290. - _R. J. Mathar_, Aug 09 2019

%e Zero ways: 169 = 13^2 cannot be equal to k * rev(k).

%e One way: 400 = 200 * 2; 10201 = 101 * 101; 162409 = 169 * 961.

%e Two ways: 7683984 = 2772 * 2772 = 1584 * 4851.

%e Three ways: 6350400 = 14400 * 441 = 25200 * 252 = 44100 * 144.

%p isA305231 := proc(n)

%p local d;

%p for d in numtheory[divisors](n) do

%p if d = digrev(n/d) then

%p return true ;

%p end if;

%p end do:

%p false ;

%p end proc:

%p n := 1;

%p for i from 0 to 4000 do

%p i2 := i^2 ;

%p if isA305231(i2) then

%p printf("%d %d\n",n,i2) ;

%p n := n+1 ;

%p end if;

%p end do: # _R. J. Mathar_, Aug 09 2019

%t {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 (Python)

%o from sympy import divisors

%o A325148_list = [0]

%o for n in range(10**6):

%o n2 = n**2

%o for m in divisors(n2):

%o if m > n:

%o break

%o if m == int(str(n2//m)[::-1]):

%o A325148_list.append(n2)

%o break # _Chai Wah Wu_, Jun 09 2021

%Y Equals A325149 Union A083408.

%Y Cf. A325149 (only one way), A083408 (at least two ways). A325150 (exactly two ways), A307019 (exactly three ways).

%Y Subsequences: A014186 (square of palindromes), A076750 (product of a non-palindrome and its reversal, where leading zeros are not allowed).

%Y Cf. A061917, A325151 (some square roots of this sequence).

%K nonn,base

%O 1,3

%A _Bernard Schott_, Apr 03 2019

%E Definition corrected by _N. J. A. Sloane_, Aug 01 2019