A306273 -id:A306273 - cp's OEIS frontend

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

Bernard Schott, Apr 03 2019

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.

			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.

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

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).

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

{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 *)

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

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

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

A307019 Squares which can be expressed as the product of a number and its reversal in exactly three different ways.

Original entry on oeis.org

6350400, 43560000, 635040000, 768398400, 4356000000, 42033200400, 55847142400, 63504000000, 64780430400, 72694944400, 76839840000, 78243278400, 234101145600, 435600000000, 4203320040000, 5086017248400, 5584714240000, 6350400000000, 6363107150400, 6478043040000, 6757504230400

Offset: 1

Bernard Schott, Mar 20 2019

1) Why do all these terms end with an even number of zeros?
1.1) Is it possible to find a term that does not end with zeros? If such a term m exists, this number must satisfy the Diophantine equation m^2 = a*rev(a) = b*rev(b) = c*rev(c). No solution (m,a,b,c) with m that does not end with zeros is known.
1.2) Consider now the Diophantine equation: m^2 = a*rev(a) = b*rev(b) where a is a palindrome and b is not a palindrome. For each solution (m,a,b), we generate terms (10*m)^2 of this sequence and we get: (10*m)^2 = 100 * m^2 = (100*a)*(rev(100*a) = (100*b)*(rev(100*b)) = (100*rev(b)) * (rev(100*rev(b))).
Example: with a(1) = 63504 = 252^2 = 252 * 252 = 144 * 441, so (m,a,b) = (63504,252,144), we obtain the 3 following ways: 6350400 = 25200 * 252 = 14400 * 441 = 44100 * 144.
2) When can square numbers be expressed in this way in more than three different ways?
If the Diophantine equation: m^2 = a*rev(a) = b*rev(b), with a <> b and a and b not palindromes has a solution, then it is possible to get integers equal to (10*m)^2 which can be expressed as the product of a number and its reversal in exactly four different ways.
We don't know if such a solution (m,a,b) exists.
David A. Corneth has found 70 terms < 6*10^15 belonging to this sequence (see links in A083408), but no square has four solutions for m^2 = k * rev(k) until 6*10^15.
There is no square less than 10^24 with 4 or more different ways. - Chai Wah Wu, Apr 12 2019

			6350400 = 2520^2 = 25200 * 252 = 14400 * 441 = 44100 * 144.
43560000 = 6600^2 = 660000 * 66 = 52800 * 825 = 82500 * 528.

Chai Wah Wu, Table of n, a(n) for n = 1..826 (n = 1..70 from David A. Corneth)
David A. Corneth, Examples of factorizations of terms as described in Name.
Shyam Sunder Gupta, Equal Product Of Reversible Numbers

Subsequence of A083406 and A083408.

Cf. A002113, A004086, A306273, A322835.

is(n) = {if(!issquare(n), return(0)); my(d = divisors(n), t = 0); forstep(i = #d, #d \ 2 + 1, -1, revd = fromdigits(Vecrev(digits(d[i]))); if(revd * d[i] == n, t++; if(t > 3, return(0)); ) ); t==3 } \\ David A. Corneth, Mar 20 2019

Corrected and extended by David A. Corneth, Mar 20 2019

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

A371034 For n >= 1, a(n) = A004086(n) if A055483(n) = 1, otherwise a(n) = n / A055483(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 31, 41, 5, 61, 71, 2, 91, 10, 7, 1, 32, 4, 52, 13, 3, 14, 92, 10, 13, 23, 1, 43, 53, 4, 73, 83, 13, 10, 14, 7, 34, 1, 5, 23, 74, 4, 94, 10, 17, 25, 35, 6, 1, 65, 19, 85, 95, 10, 16, 31, 7, 32, 56, 1, 76, 34, 23, 10, 17, 8, 37, 47

Offset: 1

Ctibor O. Zizka, Mar 31 2024

Also a(n) = R(n) if (n, R(n)) are coprime, otherwise a(n) = n / GCD(n, R(n)), where R(n) is the digit reversal of n. a(n) = 1 for n from the union of A011557 and A002113 and A001232 and A008918.

			n = 13: A004086(13) = 31, A055483(13) = 1 thus a(13) = 31.
n = 15: A004086(15) = 51, A055483(15) = 3 thus a(15) = 15/3 = 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001232, A002113, A002275, A004086, A008918, A011557, A055483, A306273.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
f:= proc(n) local r,g;
  r:= rev(n);
  g:= igcd(n,r);
  if g = 1 then r else n/g fi
end proc;
map(f, [$1..100]); # Robert Israel, Jul 09 2024

a[n_] := Module[{r = IntegerReverse[n], g}, g = GCD[n, r]; If[g == 1, r, n/g]]; Array[a, 100] (* Amiram Eldar, Mar 31 2024 *)

a(A011557(k)) = 1, k >= 0.
a(A002113(k)) = 1, k >= 2.
a(A001232(k)) = 1, k >= 1.
a(A008918(k)) = 1, k >= 1.

A325152 Numbers whose squares can be expressed as the product of a number and its reversal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 110, 111, 121, 131, 141, 151, 161, 171, 181, 191, 200, 202, 212, 220, 222, 232, 242, 252, 262, 272, 282, 292, 300, 303, 313, 323, 330, 333, 343, 353, 363, 373, 383, 393, 400, 403, 404, 414, 424, 434

Offset: 1

Bernard Schott, Apr 11 2019

The corresponding squares are in A325148 and the numbers k such that k * rev(k) is a square are in A306273.
The squares of the first 47 terms of this sequence (from 0 to 242) can be expressed as the product of a number and its reversal in only one way; then a(48) = 252 and 252^2 = 252 * 252 = 144 * 441.
The first 65 terms of this sequence (from 0 to 400) are exactly the first 65 terms of A061917; then a(66) = 403, non-palindrome, is the first term of the sequence A325151.

			One way: 20^2 = 400 = 200 * 2.
Two ways: 2772^2 = 7683984 = 2772 * 2772 = 1584 * 4851.
Three ways: 2520^2 = 14400 * 441 = 25200 * 252 = 44100 * 144.
403 is a member since 403^2 = 162409 = 169*961 (note that 403 is not a member of A281625).

Cf. A325148, A325149, A083408, A325150, A307019.
Cf. also A061917, A325151.
Similar to but different from A281625.

a(n) = sqrt(A325148(n)).

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

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A307019 Squares which can be expressed as the product of a number and its reversal in exactly three different ways.

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A371034 For n >= 1, a(n) = A004086(n) if A055483(n) = 1, otherwise a(n) = n / A055483(n).

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A325152 Numbers whose squares can be expressed as the product of a number and its reversal.

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