A083406 -id:A083406 - 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

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

A083407 Odd squares which can be expressed as the product of a number and its reversal in at least two different ways.

Original entry on oeis.org

1239016098321, 1503617940841, 1842890415961, 11151144884889, 12909311260209, 149920947896841, 181937770841761, 12128839583882121, 12598313930168521, 12639203218972521, 13081277143774921, 14462953695004441, 14999934809556841, 15260975534573041

Offset: 1

Shyam Sunder Gupta, Jun 07 2003

			1239016098321 = 1113111 * 1113111 = 1022121 * 1212201, etc.

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

Chai Wah Wu, Table of n, a(n) for n = 1..126
Shyam Sunder Gupta, EPRN Numbers.

Cf. A066531, A083406 (even squares version), A083408 (all squares version).

Corrected by Hans Havermann, Feb 13 2012
a(4)-a(14) (including four found by Hans Havermann) from Donovan Johnson, Feb 18 2012
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

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A083407 Odd squares which can be expressed as the product of a number and its reversal in at least two different ways.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI