A083407 -id:A083407 - 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

A083406 Even 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

Offset: 1

Shyam Sunder Gupta, Jun 07 2003

For n=1..49 identical to A083408.

			63504 = 252 * 252 = 144 * 441, 7683984 = 2772 * 2772 = 1584 * 4851, etc.

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

Chai Wah Wu, Table of n, a(n) for n = 1..2372 (first 73 terms from Hans Havermann)
Shyam Sunder Gupta, Equal Product of Reversible Numbers

Cf. A031877, A066531, A083407 (odd squares version), A083408 (all squares version).

Corrected and extended by Hans Havermann, Feb 11 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

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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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A083406 Even squares which can be expressed as the product of a number and its reversal in at least two different ways.

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

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