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

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 400, 484, 900, 1089, 1600, 1936, 2500, 3025, 3600, 4356, 4900, 5929, 6400, 7744, 8100, 9801, 10000, 10201, 12100, 12321, 14641, 17161, 19881, 22801, 25921, 29241, 32761, 36481, 40000, 40804, 44944, 48400, 49284, 53824, 58564, 63504, 68644, 73984, 79524, 85264

Offset: 1

Author

Bernard Schott, Apr 03 2019

Keywords

• nonn
• base

Comments

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

The squares of palindromes of A014186 are a subsequence.

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.

Examples

			Zero ways: 169 = 13^2 cannot be equal to k * rev(k).
One way: 400 = 200 * 2; 10201 = 101 * 101; 162409 = 169 * 961.
Two ways: 7683984 = 2772 * 2772 = 1584 * 4851.
Three ways: 6350400 = 14400 * 441 = 25200 * 252 = 44100 * 144.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms for n = 1..231 from R. J. Mathar)
• Bernard Schott, The different ways

Crossrefs

Equals A325149 Union A083408.

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

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

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

Programs

Maple

isA305231 := proc(n)
    local d;
    for d in numtheory[divisors](n) do
        if d = digrev(n/d) then
            return true ;
        end if;
    end do:
    false ;
end proc:
n := 1;
for i from 0 to 4000 do
    i2 := i^2 ;
    if isA305231(i2) then
        printf("%d %d\n",n,i2) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019

Mathematica

{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 *)

Python

from sympy import divisors
A325148_list = [0]
for n in range(10**6):
    n2 = n**2
    for m in divisors(n2):
        if m > n:
            break
        if m == int(str(n2//m)[::-1]):
            A325148_list.append(n2)
            break # Chai Wah Wu, Jun 09 2021

Formula

Intersection of A305231 and A000290. - R. J. Mathar, Aug 09 2019

Extensions

Definition corrected by N. J. A. Sloane, Aug 01 2019