A014186 -id:A014186 - 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

A325149 Squares which can be expressed as the product of a number and its reverse in exactly one way.

Original entry on oeis.org

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, 68644, 73984, 79524, 85264, 90000

Offset: 1

Bernard Schott, Apr 03 2019

The first 47 terms of this sequence (from 0 to 58564) are identical to the first 47 terms of A325148. The square 63504 is not present because it can be expressed in two ways: 63504 = 252 * 252 = 144 * 441.
There are three families of squares in this sequence:
1) Squares of palindromes in A002113\A117281.
2) Squares of non-palindromes which form the sequence A325151.
These squares are a subsequence of A076750.
3) Squares of (m*10^q) with q >= 1 and m palindrome in A002113\A117281.

			For each family:
1) Square of palindromes: 53824 = 232^2 = 232 * 232.
2) Square of non-palindromes m^2 = k*rev(k) with k and rev(k) which have the same number of digits: 162409 = 403^2 = 169 * 961.
3) Square ends with zeros: 48400 = 220^2 = 2200 * 22.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition, p. 168.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A325148 (at least one way), A083408 (at least two ways), A325150 (exactly two ways), A307019 (exactly three ways).

Cf. A014186 (squares of palindromes), A076750.

a(52) corrected by Chai Wah Wu, Apr 11 2019

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

A115743 Squares that are the product of 2 palindromes greater than 1.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 64, 81, 121, 484, 1089, 1764, 1936, 2401, 2704, 3025, 4356, 5929, 6084, 7744, 9801, 10201, 12321, 14641, 17161, 19881, 22801, 25921, 27225, 29241, 32761, 36481, 40804, 44944, 49284, 53824, 58564, 63504, 68644, 69696

Offset: 1

Giovanni Resta, Jan 31 2006

Are most terms of the form p^2 where p is a palindrome? - David A. Corneth, May 25 2021

			6084 = 78^2 and 6084 = 9*676.

David A. Corneth, Table of n, a(n) for n = 1..13971 (terms <= 10^15)

Cf. A014186, A115683, A115744, A115745.

With[{nn=50000},Select[Union[Select[Times@@@Tuples[Select[Range[2,nn],PalindromeQ],2],IntegerQ[ Sqrt[ #]]&]],#<=2 nn&]] (* Harvey P. Dale, Aug 21 2022 *)

A014187 Cubes of palindromes.

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1331, 10648, 35937, 85184, 166375, 287496, 456533, 681472, 970299, 1030301, 1367631, 1771561, 2248091, 2803221, 3442951, 4173281, 5000211, 5929741, 6967871

Offset: 0

N. J. A. Sloane

Cf. A002113, A014186.

Select[Range[0,200],PalindromeQ]^3 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 02 2017 *)

A014188 Fourth powers of palindromes.

Original entry on oeis.org

0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 14641, 234256, 1185921, 3748096, 9150625, 18974736, 35153041, 59969536, 96059601, 104060401, 151807041, 214358881, 294499921, 395254161, 519885601, 671898241

Offset: 0

N. J. A. Sloane

Cf. A002113, A014186.

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]; Select[ Range[ 0,200],palQ]^4 (* Harvey P. Dale, Feb 09 2015 *)

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

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A325149 Squares which can be expressed as the product of a number and its reverse in exactly one way.

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A115743 Squares that are the product of 2 palindromes greater than 1.

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A014187 Cubes of palindromes.

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A014188 Fourth powers of palindromes.

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