A035090 -id:A035090 - cp's OEIS frontend

A035123 Roots of 'non-palindromic squares remaining square when written backwards'.

12, 13, 21, 31, 33, 99, 102, 103, 112, 113, 122, 201, 211, 221, 301, 311, 1002, 1003, 1011, 1012, 1013, 1021, 1022, 1031, 1101, 1102, 1103, 1112, 1113, 1121, 1122, 1201, 1202, 1211, 1212, 1301, 2001, 2011, 2012, 2021, 2022, 2101, 2102, 2111, 2121

Offset: 1

Patrick De Geest, Nov 15 1998

Those with trailing zeros are excluded.

Union of ordered entries of A106323 and A106324. - Lekraj Beedassy, May 02 2005

			1212^2 = 1468944 -> 4498641 = 2121^2.

Jon E. Schoenfield, Table of n, a(n) for n = 1..8448 (first 798 terms from Seiichi Manyama; all terms < 10^10)
Patrick De Geest, Palindromic Squares in bases 2 to 17
Eric Weisstein's World of Mathematics, Square Number

Cf. A035090, A035125.

r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Select[Range[2200], Mod[#, 10] != 0 && IntegerQ[Sqrt[r[#^2]]] && r[#^2] != #^2 &] (* Jean-François Alcover, Mar 08 2011 *)

A319388 Non-palindromic squares.

Original entry on oeis.org

16, 25, 36, 49, 64, 81, 100, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 529, 576, 625, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025

Offset: 1

Seiichi Manyama, Sep 18 2018

Intersection of A000290 and A029742. - Felix Fröhlich, Sep 18 2018

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000290, A002113, A002779, A029742, A035090, A319389.

[n^2: n in [0..60] | not Intseq(n^2) eq Reverse(Intseq(n^2))]; // Vincenzo Librandi, Sep 19 2018

ispali:= proc(n) local L;
L:= convert(n,base,10);
L = ListTools:-Reverse(L)
end proc:
remove(ispali, [seq(i^2,i=1..100)]); # Robert Israel, Sep 18 2018

 pb10Q[n_]:=!Module[{idn10=IntegerDigits[n, 10]}, idn10==Reverse[idn10]]; Select[Range[0, 3100]^2, pb10Q] (* Vincenzo Librandi, Sep 19 2018 *)

terms(n) = my(i=0); for(k=0, oo, if(i==n, break); my(s=k^2, d=digits(s)); if(d!=Vecrev(d), print1(s, ", "); i++))
/* Print initial 50 terms as follows */
terms(50) \\ Felix Fröhlich, Sep 18 2018

A322835 Non-palindromic numbers n such that n * reverse(n) is a square and n and reverse(n) do not have the same number of digits.

Original entry on oeis.org

100, 200, 300, 400, 500, 600, 700, 800, 900, 1100, 2200, 3300, 4400, 5500, 6600, 7700, 8800, 9900, 10000, 10100, 11100, 12100, 13100, 14100, 14400, 15100, 16100, 16900, 17100, 18100, 19100, 20000, 20200, 21200, 22200, 23200, 24200, 25200, 26200, 27200, 28200, 28800, 29200, 30000, 30300

Offset: 1

Bernard Schott, Jan 02 2019

The terms in this sequence are mostly of the form m * 100^k with k >= 1, but this condition is not sufficient.
A062917 U {this sequence} = A070760, with empty intersection.
There are exactly four families of such integers here: numbers of the forms A002113(j)*100^k, A035090(j)*100^k, A082994(j)*100^k and A323061(j)*10^(2k+1).
All terms are multiples of 10, but they are not necessarily multiples of 100. The first multiple of 10 that is not a multiple of 100 is a(755) = 5449680, and there are only 30 such terms among the first 10000 terms. - Chai Wah Wu, Jan 07 2019

			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. - _Chai Wah Wu_, Jan 07 2019

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002113, A004086, A035090, A062917, A070760, A082994, A323061.

Select[100 Range@303, And[! PalindromeQ@ #, IntegerQ@ Sqrt[#1 #2], UnsameQ @@ IntegerLength@ {#1, #2}] & @@ {#, IntegerReverse@ #} &] (* Michael De Vlieger, Jan 03 2019 *)

is(n) = n % 10 == 0 && issquare(n * fromdigits(Vecrev(digits(n)))) \\ David A. Corneth, Jan 03 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

A035124 Nonpalindromic cubes remaining cubic which written backwards: take the cube root of n, reverse its digits, cube that and the result is n with its digits reversed.

Original entry on oeis.org

1033364331, 1334633301, 1003303631331, 1331363033001, 1000330036301331, 1003033061330301, 1003333697667631, 1030331603303001, 1030637669664331, 1331036300330001, 1334669667360301, 1367667963333001, 1000033000363001331, 1000303030604030301, 1000333036964367631

Offset: 1

Patrick De Geest, Nov 15 1998

Cubes with trailing zeros are excluded.

			1011^3 = 1033364331 -> 1334633301 = 1101^3.
1003333697667631 is included because its cube root, 100111, when reversed (i.e., 111001) and cubed yields 1367667963333001.

P. De Geest, Palindromic Cubes
Eric Weisstein's World of Mathematics, Cubic Number
Index entry for sequences related to reversing digits of squares

Cf. A035090, A035125, A064021, A319389.

isok(n) = {if (ispower(n, 3, &k), dn = digits(n); if (Vecrev(dn) != dn, dk = Vecrev(digits(k)); rk = subst(Pol(dk, x), x, 10); digits(rk^3) == Vecrev(dn);););} \\ Michel Marcus, Oct 04 2015

More terms from Seiichi Manyama, Sep 18 2018

A035125 Roots of 'non-palindromic cubes remaining cubic when written backwards'.

Original entry on oeis.org

1011, 1101, 10011, 11001, 100011, 100101, 100111, 101001, 101011, 110001, 110101, 111001, 1000011, 1000101, 1000111, 1001011, 1001101, 1010001, 1010011, 1011001, 1100001, 1100101, 1101001, 1110001, 10000011, 10000101, 10000111, 10001001, 10001011, 10001101, 10010001

Offset: 1

Patrick De Geest, Nov 15 1998

Those with trailing zeros are excluded. Binary look is fortuitous!

			1000011^3 = 1000033000363001331 -> 1331003630003300001 = 1100001^3.

P. De Geest, Palindromic Cubes
Eric Weisstein's World of Mathematics, Cubic Number
Index entry for sequences related to reversing digits of squares

Cf. A035090, A035124.

More terms from Seiichi Manyama, Sep 18 2018

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

A064021 Squares k^2 such that reverse(k)^2 = reverse(k^2), excluding squares of palindromes.

Original entry on oeis.org

144, 169, 441, 961, 10404, 10609, 12544, 12769, 14884, 40401, 44521, 48841, 90601, 96721, 1004004, 1006009, 1022121, 1024144, 1026169, 1042441, 1044484, 1062961, 1212201, 1214404, 1216609, 1236544, 1238769, 1256641, 1258884, 1442401

Offset: 1

Harvey P. Dale, Sep 18 2001

Subsequence of A035090. - M. F. Hasler, Mar 22 2011

			1026169 is included because its square root, 1013, when reversed (i.e., 3101) and squared yields 9616201.
Squares < 10 and 121 = 11^2, 484 = 22^2, ... are not in the sequence, since they are the square of a palindrome. - _M. F. Hasler_, Mar 22 2011

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 124, 127 (Rev. ed. 1997).

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 750 terms from Harry J. Smith)

Cf. A035124, A035123, A033294, A140212.

Cases[Range[2000]^2, k_ /; Mod[k, 10] != 0 && IntegerDigits[k] != Reverse[IntegerDigits[k]] && FromDigits[Reverse[IntegerDigits[Sqrt[k]]]]^2 == FromDigits[Reverse[IntegerDigits[k]]]] (* Jean-François Alcover, Mar 22 2011 *)
Select[Range[1250]^2,!PalindromeQ[Sqrt[#]]&&IntegerReverse[#] == IntegerReverse[ Sqrt[#]]^2 &&Mod[#,10]!=0&] (* Harvey P. Dale, Jul 01 2022 *)

Rev(x)= { local(d,r); r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); return(r) }
{ n=0; for (m=1, 10^9, if (m%10==0, next); x=m^2; r=Rev(x); if (r==x, next); if (r==Rev(m)^2, write("b064021.txt", n++, " ", x); if (n==750, break)) ) } \\ Harry J. Smith, Sep 06 2009

{n = A000290(k) such that A004086(A000290(k)) = A000290(A004086(k)) and k is not in A002113}. - Jonathan Vos Post, May 02 2011

a(n) = A140212(n)^2. - Giovanni Resta, Jun 22 2018

A106323 Smaller of number pair whose squares are reversals of each other, with no leading zeros allowed.

Original entry on oeis.org

12, 13, 33, 102, 103, 112, 113, 122, 1002, 1003, 1011, 1012, 1013, 1021, 1022, 1031, 1102, 1103, 1112, 1113, 1121, 1122, 1202, 1212, 2012, 2022, 3168, 10002, 10003, 10011, 10012, 10013, 10021, 10022, 10031, 10102, 10103, 10111, 10112, 10113

Offset: 1

Lekraj Beedassy, Apr 29 2005

For numbers whose squares are the reversal of a(n)^2, see A106324.

			33 is in the sequence because 33^2=1089 and we have 9801=99^2. Likewise,122^2=14884 and we have 48841=221^2.

Cf. A035090; A035123.

Half of A035123.

isok(n) = {if (n % 10 == 0, return (0)); d = digits(n^2, 10); m = sum(k=0, #d-1, d[k+1]*10^(k)); if (! issquare(m), return (0)); return (n < sqrtint(m));} \\ Michel Marcus, Jul 28 2013

Corrected and extended by Joshua Zucker, May 12 2006

cp's OEIS Frontend

A035123 Roots of 'non-palindromic squares remaining square when written backwards'.

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A319388 Non-palindromic squares.

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A322835 Non-palindromic numbers n such that n * reverse(n) is a square and n and reverse(n) do not have the same number of digits.

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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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A035124 Nonpalindromic cubes remaining cubic which written backwards: take the cube root of n, reverse its digits, cube that and the result is n with its digits reversed.

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A035125 Roots of 'non-palindromic cubes remaining cubic when written backwards'.

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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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