A117281 -id:A117281 - cp's OEIS frontend

A083408 Squares which can be expressed as the product of a number and its reversal in at least two different ways.

63504, 435600, 6350400, 7683984, 16240900, 25401600, 43560000, 66585600, 420332004, 558471424, 635040000, 647804304, 726949444, 768398400, 782432784, 1067328900, 1624090000, 1897473600, 2341011456, 2540160000, 4356000000, 6658560000, 42033200400, 50860172484, 52587662400

Offset: 1

Shyam Sunder Gupta, Jun 07 2003

Union of A083406 and A083407. - Lekraj Beedassy, Apr 23 2006

			63504 = 252 * 252 = 144 * 441,
1239016098321 = 1113111 * 1113111 = 1022121 * 1212201, etc.
635040000 = 144 * 4410000 = 252 * 2520000 = 441 * 1440000. - _David A. Corneth_, Mar 22 2019

S. S. Gupta, EPRNs, Science Today, Feb. 1987, India.

Chai Wah Wu, Table of n, a(n) for n = 1..2498 (n = 1..76 from Hans Havermann, n = 77..241 from David A. Corneth)
David A. Corneth, Examples of factorizations of terms as described in Name.
Shyam Sunder Gupta, EPRN Numbers.
Shyam Sunder Gupta, Equal Product of Reversible Numbers (EPRN), Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 12, 353-365.

Cf. A062917, A066531, A083406 (even), A083407 (odd), A070760, A117281 (palindromic square roots), A206642 (non-palindromic square roots), A325150 (products in exactly two different ways), A307019 (products in exactly three different ways).

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 >= 2, return(1)); ) ); 0 } \\ David A. Corneth, Mar 21 2019

Corrected and extended by Hans Havermann, Feb 11 2012
a(21)-a(25) from David A. Corneth, Mar 21 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

A306273 Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 101, 111, 121, 131, 141, 144, 151, 161, 169, 171, 181, 191, 200, 202, 212, 222, 232, 242, 252, 262, 272, 282, 288, 292, 300, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 400, 404, 414, 424, 434, 441, 444, 454, 464, 474, 484, 494, 500, 505, 515, 525, 528, 535

Offset: 1

Bernard Schott, Feb 02 2019

The first nineteen terms are palindromes (cf. A002113). There are exactly seven different families of integers which together partition the terms of this sequence. See the file "Sequences and families" for more details, comments, formulas and examples.
From Chai Wah Wu, Feb 18 2019: (Start)
If w is a term with decimal representation a, then the number n corresponding to the string axa is also a term, where x is a string of k repeated digits 0 where k >= 0. The number n = w*10^(k+m)+w = w*(10^(k+m)+1) where m is the number of digits of w. Then R(n) = R(w)*10^(k+m)+R(w) = R(w)(10^(k+m)+1). Then n*R(n) = w*R(w)(10^(k+m)+1)^2 which is a square since w is a term.
The same argument shows that numbers corresponding to axaxa, axaxaxa, ... are also terms.
For example, since 528 is a term, so are 528528, 5280528, 52800528, 5280052800528, etc.
(End)

			One example for each family:
family 1 is A002113: 323 * 323 = 323^2;
family 2 is A035090: 169 * 961 = 13^2 * 31^2 = 403^2;
family 3 is A082994: 288 * 882 = (2*144) * (2*441) = 504^2;
family 4 is A002113(j) * 100^k: 75700 * 757 = 7570^2;
family 5 is A035090(j) * 100^k: 44100 * 144 = 2520^2;
family 6 is A082994(j) * 100^k: 8670000 * 768 = 81600^2;
family 7 is A323061(j) * 10^(2k+1): 5476580 * 856745 = 2166110^2.

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition (1997), p. 168.

Robert Israel, Table of n, a(n) for n = 1..10000
Bernard Schott, Sequences and Families
Eric Weisstein's World of Mathematics, Reversal

Cf. A002113, A070760, A062917, A035090, A082994, A322835, A323061.

Cf. A083406, A083407, A083408, A117281 (Squares = k * rev(k) in at least two ways).

revdigs:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= n -> issqr(n*revdigs(n)):
select(filter, [$0..1000]);# Robert Israel, Feb 09 2019

Select[Range[0, 535], IntegerQ@ Sqrt[# IntegerReverse@ #] &] (* Michael De Vlieger, Feb 03 2019 *)

isok(n) = issquare(n*fromdigits(Vecrev(digits(n)))); \\ Michel Marcus, Feb 04 2019

A206642 Non-palindromes whose squares are in A066531.

Original entry on oeis.org

660, 2520, 4030, 5040, 6600, 8160, 25200, 27720, 32670, 40300, 43560, 50400, 66000, 81600, 205020, 229320, 236320, 254520, 269620, 277200, 279720, 310030, 326700, 329670, 351430, 410040, 435600, 439560, 458640, 483840, 486160

Offset: 1

Hans Havermann, Feb 11 2012

			660 is in the sequence because 660^2 = 528*825 = 6600*0066; 2520 is in the sequence because 2520^2 = 14400*00441 = 25200*00252 = 44100*00144; etc.

Shyam Sunder Gupta, EPRNs

Cf. A066531, A083408 (square EPRNs), A117281 (palindromic version).

Edited by N. J. A. Sloane, Aug 01 2019

A258382 Non-palindromic numbers n such that the square root of n multiplied by the reversal of n is a palindrome.

Original entry on oeis.org

144, 441, 1584, 4851, 10404, 12544, 14544, 14884, 15984, 27648, 40401, 44521, 44541, 48841, 48951, 84672, 114444, 137984, 144144, 159984, 409739, 441441, 444411, 489731, 489951, 937904, 1004004, 1022121, 1024144, 1042441, 1044484, 1050804

Offset: 1

Pieter Post, May 28 2015

This sequence is infinite, because it contains several infinite subsequences such as: sqrt(1584*4851)=2772, sqrt(15984*48951)=27972, sqrt(159..984*489...951)=279...972.

It appears that the first (or last) digit is never 5, 6 or 7.

			27648 is in the sequence because sqrt(27648*84672)=48384.

Pieter Post and Giovanni Resta, Table of n, a(n) for n = 1..1124 (first 246 terms from Pieter Post, terms < 10^12)

Cf. A002113, A002778, A059744, A004086, A117281.

palQ[n_] := Block[{d = IntegerDigits@ n}, And[IntegerQ@ n, d == Reverse@ d]]; Select[Range@ 100000, And[! palQ@ #, palQ[Sqrt[# FromDigits@ Reverse@ IntegerDigits@ #]]] &] (* Michael De Vlieger, May 28 2015 *)

rev(k) = subst(Polrev(digits(k)), x, 10);
isok(n) = {rn = rev(n); if (rn != n, nrn = n*rn; issquare(nrn) && (y=sqrtint(nrn)) && (y == rev(y)););} \\ Michel Marcus, May 29 2015

for n in range (1, 10**9):
    y=int(str(n)[::-1])
    ya=int(pow(n*y,1/2))
    if ya==int(str(ya)[::-1]) and n*y==ya**2 and n!=y:
        print (n)

Numbers n such that sqrt(n*reversal(n)) is a palindrome, where n is not a palindrome.

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

cp's OEIS Frontend

A083408 Squares which can be expressed as the product of a number and its reversal in at least two different ways.

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

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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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A306273 Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal.

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A206642 Non-palindromes whose squares are in A066531.

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A258382 Non-palindromic numbers n such that the square root of n multiplied by the reversal of n is a palindrome.

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

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