A322835 -id:A322835 - cp's OEIS frontend

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

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

A070760 Numbers k such that k*rev(k) is a square different from k^2, where rev=A004086, decimal reversal.

Original entry on oeis.org

100, 144, 169, 200, 288, 300, 400, 441, 500, 528, 600, 700, 768, 800, 825, 867, 882, 900, 961, 1089, 1100, 1584, 2178, 2200, 3300, 4400, 4851, 5500, 6600, 7700, 8712, 8800, 9801, 9900, 10000, 10100, 10404, 10609, 10989

Offset: 1

Reinhard Zumkeller, May 15 2002

If k is a palindrome (A002113), then 100*k is a term. If k is a term, then 100*k is a term. - Chai Wah Wu, Mar 31 2018
From Bernard Schott, Jan 02-10 2019: (Start)
There are six different families of integers in this sequence.
1) If k and rev(k) do not have the same number of digits:
All these integers are in A322835 where the first four families are explained and detailed.
Family 1: A002113(j) * 100^k
Family 2: A035090(j) * 100^k
Family 3: A082994(j) * 100^k
Family 4: A323061(j) * 10^(2k+1)
2) If k and rev(k) have the same number of digits.
All these integers are in A062917.
Family 5: Non-palindromic squares whose reverse is also square. These integers are in A035090.
Family 6: Non-palindromic numbers k, such that k * rev(k) is a square, with k and rev(k) not both square. These integers are in A082994.
3) Relationships between these different sequences.
  A035090 Union A082994 = A062917 with empty intersection, and,
A062917 Union A322835 = {This sequence} with empty intersection. (End)

			a(2)=144: rev(144)=441, 144*441=(12^2)*(21^2)=(12*21)^2 and 144<>12*21=252.
From _Bernard Schott_, Jan 02 2019: (Start)
Example for family 1: 200 * 2 = 400 = 20^2
Example for family 2: 14400 * 441 = 120^2 * 21^2 = 2520^2
Example for family 3: 28800 * 882 = (2 * 120^2) * (2 * 21^2) = 5040^2
Example for family 4: 5449680 * 869445 = 2176740^2
Example for family 5: 169 * 961 = 13^2 * 31^2 = 403^2
Example for family 6: 528 * 825 = (33 * 4^2) * (33 * 5^2) = 660^2. (End)

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Reinhard Zumkeller)

Cf. A002113, A061205, A010052, A035090, A062917, A082994, A322835, A323061.

a070760 n = a070760_list !! (n-1)
a070760_list = [x | x <- [0..], let y = a061205 x,
                    y /= x ^ 2, a010052 y == 1]
-- Reinhard Zumkeller, Apr 10 2012, Apr 29 2011

Select[ Range[11000], (k = Sqrt[ # * FromDigits @ Reverse @ IntegerDigits[#]]; IntegerQ[k] && k != #) &] (* Jean-François Alcover, Nov 30 2011 *)
sdnQ[n_]:=Module[{c=n*IntegerReverse[n]},c!=n^2&&IntegerQ[Sqrt[c]]]; Select[ Range[11000],sdnQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 25 2016 *)

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

A082994 Numbers n such that all the following properties hold: (i) n*reverse(n) is a square; (ii) n != reverse(n); (iii) n and reverse(n) are not both squares; and (iv) n and reverse(n) have the same number of digits.

Original entry on oeis.org

288, 528, 768, 825, 867, 882, 1584, 2178, 4851, 8712, 10989, 13104, 14544, 15984, 20808, 21978, 26208, 27648, 27848, 36828, 40131, 44541, 48139, 48951, 49686, 57399, 68694, 80262, 80802, 82863, 84672, 84872, 87912, 93184, 98901, 99375

Offset: 1

Jason Earls, May 29 2003

These numbers are counterexamples to the following conjecture given in the Ogilvy-Anderson reference: "When an integer and its reversal are unequal, their product is never a square except when both are squares." This sequence excludes terms like 2200, i.e. 2200*22 = 48400.
Contains x*(10^k+1) for k >= 3 with x in {144, 169, 288, 441, 528, 768, 825, 867, 882, 961}. - Robert Israel, Jun 11 2018
A035090 U {this sequence} = A062917, with empty intersection. - Bernard Schott, Jan 04 2019

			a(5) = 867 because 867 * 768 = 665856 = 816^2.

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.
J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 82-83. ASIN: B002ACVZ6O [From Jason Earls, Nov 22 2009]

Robert Israel, Table of n, a(n) for n = 1..168
M. A. Rashid, M. A. Uppal, D. C. B. Marsh and A. Wayne, Product of a Number and Its Reverse, American Mathematical Monthly, vol. 64 (1957), p. 434, E-1243. - _Felix Fröhlich_, Jul 11 2014

Cf. A002113, A004086, A035090, A062917, A070760, A322835.

revdigs:= proc(n) local L;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(n) local r;
  if issqr(n) then return false fi;
  r:= revdigs(n);
  r <> n and issqr(r*n) and not issqr(r);
end proc:
select(filter, [seq(seq(10*i+j,j=1..9),i=1..10^4)]); # Robert Israel, Jun 11 2018

Select[Range[10^5], And[UnsameQ @@ {#1, #2}, IntegerQ@ Sqrt[#1 #2], AllTrue[{#1, #2}, ! IntegerQ@ Sqrt@ # &], SameQ @@ (IntegerLength@ {#1, #2})] & @@ {#, IntegerReverse@ #} &] (* Michael De Vlieger, Jan 04 2019 *)

Name clarified by Bernard Schott, Jan 04 2019

A323061 Numbers n that are not multiples of 10 and such that 10nR(n) is a square, where R(n) = A004086(n) is the decimal digits of n reversed.

Original entry on oeis.org

544968, 547658, 560106, 601065, 856745, 869445, 2495295, 4601685, 5606106, 5861064, 5925942, 6016065, 20861005, 21778875, 22972005, 29389855, 42251835, 50016802, 50027922, 51826326, 53815224, 55898392, 56066106, 56570706, 56873466, 57887712, 60166065, 60707565

Offset: 1

Chai Wah Wu, Jan 07 2019

10*a(n) are exactly the terms in A322835 that are not multiples of 100.
m is a term if and only if R(m) is a term.
The product of the first and last digits of a term is a multiple of 10, i.e., the first and last digits of a term are the digit 5 and an even nonzero digit.
The sequence has an infinite number of terms. For instance, 601x065 is a term where x is a string of k repeated digits 6 and k >= 0, i.e., 601065, 6016065, 60166065, etc. Similarly numbers of the form 560x106 are also terms.
To see this, let a = 601*10^(3+k) + 65 + 6000*(10^k-1)/9. Then R(a) = 56*10^(4+k) + 106 + 6000*(10^k-1)/9. The number 10*a*R(a) can be written as 30360100*(10^(k + 3) - 1)^2/9 whose square root is 5510*(10^(k + 3) - 1)/3.
From Chai Wah Wu, Feb 18 2019: (Start)
22994x77005 is a term where x is a string of k repeated digits 9 and k >= 0. Let a = 22994*10^(5+k) + 77005 + 100000*(10^k-1). Then R(a) = 50077*10^(5+k) + 49922 + 100000*(10^k-1). The number 10*a*R(a) can be written as 11515436100*(10^(k+5) - 1)^2, whose square root is 107310*(10^(k+5) - 1).
23804x76195 is a term where x is a string of k repeated digits 9 and k >= 0. Let a = 23804*10^(5+k) + 76195 + 100000*(10^k-1). Then R(a) = 59167*10^(5+k) + 40832 + 100000*(10^k-1). The number 10*a*R(a) can be written as 14084942400*(10^(k+5) - 1)^2, whose square root is 118680*(10^(k+5) - 1).
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 and k >= 0. The number n = w*10^(k+m)+w = w*(10^(k+m)+1) where m is the number of digits of w. Note that n is also not a multiple of 10. Then R(n) = R(w)*10^(k+m)+R(w) = R(w)(10^(k+m)+1). Then 10*n*R(n) = 10*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 544968 is a term, so are 544968544968, 5449680544968, 54496800544968, 5449680054496800544968, etc.
(End)

			238026195 * 591620832 * 10 = 1186681320^2.

Giovanni Resta, Table of n, a(n) for n = 1..200 (first 54 terms from Chai Wah Wu)

Cf. A004086, A322835.

Select[Range[61*10^6],Mod[#,10]!=0&&IntegerQ[Sqrt[10# IntegerReverse[ #]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 16 2020 *)

isok(n) = (n % 10) && issquare(10*n*fromdigits(Vecrev(digits(n)))); \\ Michel Marcus, Jan 10 2019

cp's OEIS Frontend

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

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A070760 Numbers k such that k*rev(k) is a square different from k^2, where rev=A004086, decimal reversal.

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

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A082994 Numbers n such that all the following properties hold: (i) n*reverse(n) is a square; (ii) n != reverse(n); (iii) n and reverse(n) are not both squares; and (iv) n and reverse(n) have the same number of digits.

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A323061 Numbers n that are not multiples of 10 and such that 10*n*R(n) is a square, where R(n) = A004086(n) is the decimal digits of n reversed.

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A323061 Numbers n that are not multiples of 10 and such that 10nR(n) is a square, where R(n) = A004086(n) is the decimal digits of n reversed.