A325151 -id:A325151 - 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

A325150 Squares which can be expressed as the product of a number and its reversal in exactly two ways.

Original entry on oeis.org

63504, 435600, 7683984, 16240900, 25401600, 66585600, 420332004, 558471424, 647804304, 726949444, 782432784, 1067328900, 1624090000, 1897473600, 2341011456, 2540160000, 6658560000, 50860172484, 52587662400, 63631071504, 67575042304, 78384320784, 96118600900, 106732890000

Offset: 1

Bernard Schott, Apr 03 2019

When q = m^2 does not end with a 0 is a term, then m is a palindrome belonging to A117281.

When q = m^2 ending with a 0 is a term, then either m = r * 10^u where r belongs to A325151 and u >= 1, or m is in A342994.

			1) Squares without trailing zeros:
Even square: 7683984 = 2772^2 = 2772 * 2772 = 1584 * 4851.
Odd square: 1239016098321 = 1113111^2 = 1113111 * 1113111 = 1022121 * 1212201.
2) Squares with trailing zeros:
1st case: 16240900 = 4030^2 = 16900 * 961 = 96100 * 169.
2nd case: 435600 = 660^2 = 6600 * 66 = 528 * 825.

D. Wells, 63504 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 168.

Chai Wah Wu, Table of n, a(n) for n = 1..1672
David A. Corneth, Examples of factorizations of terms in exactly two or three ways.
Shyam Sunder Gupta, EPRN Numbers.

Cf. A325148 (at least one way), A325149 (only one way), A083408 (at least two ways), A307019 (exactly three ways).
Cf. A083407 (odd squares), A083408 (even squares without trailing 0's).
Cf. A117281, A325151, A342994.

Corrected terms by Chai Wah Wu, Apr 12 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

A342994 a(n) = (1000^n - 1)*(220/333).

Original entry on oeis.org

660, 660660, 660660660, 660660660660, 660660660660660, 660660660660660660, 660660660660660660660, 660660660660660660660660, 660660660660660660660660660, 660660660660660660660660660660, 660660660660660660660660660660660, 660660660660660660660660660660660660

Offset: 1

Bernard Schott, Apr 28 2021

Why is this sequence interesting? Answer: Squares that can be expressed as the product of a number and its reversal in exactly two ways are in A325150.
There are only 3 ways to get such squares, according to the Diophantine equation q = m^2 = k * rev(k) = t * rev(t).
1) When q, m do not end with 0, then m = k is a palindrome belonging to A117281; example: for m = k = A117281(1) = 252, q = 252^2 = 252*252 = 144*441 = 63504 = A325150(1).
2) When q = m^2 both end with 0, there exist these 2 possibilities:
2.1) k and t also both end with 0, then m = r * 10^u where r belongs to A325151 and u >= 1; example: for r = A325151(1) = 403, u = 1, m = 4030, k = 16900 and t = 96100 with q = 16240900 = 4030^2 = 16900 * 961 = 96100 * 169 = A325150(4).
2.2) k ends with 0 but not t, then m belongs to this sequence; so another equivalent name is: numbers with trailing zeros whose square can be expressed as the product of a number ending with 0 and its reversal, and agian as the product of a number and its reversal, but this time without trailing zero (see examples).

			For a(1) = 660, we have 660^2 = 435600 = 6600 * 66 = 528 * 825 = A325150(2) (q = 435600, m = 660, k = 6600, t = 528).
For a(2) = 660660, we have 660660^2 = 436471635600 = 6606600 * 66066 = 528528 * 825825 (q = 436471635600, m = 660660, k = 6606600, t = 528528).
Generalization: for a(n) = 660...660, we have 660...660^2 = 660...6600 * 660...66 = 528...528 * 825...825.

Index entries for linear recurrences with constant coefficients, signature (1001,-1000).

Cf. A117281, A325150, A325151.

E:= seq((1000^n - 1)*(220/333), n=1..11);

Table[(1000^n - 1)*(220/333), {n, 1, 11}] (* Amiram Eldar, Apr 28 2021 *)

Vec(660*x/((1000*x-1)*(x-1)) + O(x^13)) \\ Elmo R. Oliveira, Jul 01 2025

a(n) = (1000^n - 1)*(220/333).
G.f.: 660*x/(1 - 1001*x + 1000*x^2). - Stefano Spezia, Apr 28 2021
a(n) = 1001*a(n-1) - 1000*a(n-2). - Wesley Ivan Hurt, Apr 28 2021
E.g.f.: 220*exp(x)*(-1 + exp(999*x))/333. - Elmo R. Oliveira, Jul 01 2025

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)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A325150 Squares which can be expressed as the product of a number and its reversal in exactly two ways.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A342994 a(n) = (1000^n - 1)*(220/333).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula