A035123 -id:A035123 - cp's OEIS frontend

A035090 Non-palindromic squares which when written backwards remain square (and still have the same number of digits).

144, 169, 441, 961, 1089, 9801, 10404, 10609, 12544, 12769, 14884, 40401, 44521, 48841, 90601, 96721, 1004004, 1006009, 1022121, 1024144, 1026169, 1042441, 1044484, 1062961, 1212201, 1214404, 1216609, 1236544, 1238769, 1256641

Offset: 1

Patrick De Geest, Nov 15 1998

Squares with trailing zeros not included.

Sequence is infinite, since it includes, e.g., 10^(2k) + 4*10^k + 4 for all k. - Robert Israel, Sep 20 2015

Robert Israel, Table of n, a(n) for n = 1..798
Patrick De Geest, Palindromic Squares in bases 2 to 17
Index entry for sequences related to reversing digits of squares

Reversing a polytopal number gives a polytopal number:
cube to cube: A035123, A035124, A035125, A002781;
square to square: A161902, A035090, A033294, A106323, A106324, A002779;
square to triangular: A181412, A066702;
tetrahedral to tetrahedral: A006030;
triangular to square: A066703, A179889;
triangular to triangular: A066528, A069673, A003098, A066569.
Cf. A319388.

rev:= proc(n) local L,i;
L:= convert(n,base,10);
add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(n) local t;
  if n mod 10 = 0 then return false fi;
  t:= rev(n);
t <> n and issqr(t)
end proc:
select(filter, [seq(n^2, n=1..10^5)]); # Robert Israel, Sep 20 2015

Select[Range[1200]^2,!PalindromeQ[#]&&IntegerLength[#]==IntegerLength[ IntegerReverse[ #]] && IntegerQ[Sqrt[IntegerReverse[#]]]&] (* Harvey P. Dale, Jul 19 2023 *)

a(n) = A035123(n)^2. - R. J. Mathar, Jan 25 2017

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

A106324 Greater of number pair whose squares are reversals of each other, with no leading zeros allowed.

Original entry on oeis.org

21, 31, 99, 201, 301, 211, 311, 221, 2001, 3001, 1101, 2101, 3101, 1201, 2201, 1301, 2011, 3011, 2111, 3111, 1211, 2211, 2021, 2121, 2102, 2202, 6501, 20001, 30001, 11001, 21001, 31001, 12001, 22001, 13001, 20101, 30101, 11101, 21101, 31101

Offset: 1

Lekraj Beedassy, Apr 29 2005

Terms are ordered according to the smaller member of the pair (A106323). - Michel Marcus, Jul 28 2013

Cf. A035090; A035123.

Half of A035123.

Corrected and extended by Joshua Zucker, May 12 2006

A140212 Numbers n not a multiple of 10 such that reverse(n^2) = reverse(n)^2, but reverse(n) is different from n.

Original entry on oeis.org

12, 13, 21, 31, 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, 2201, 2202, 2211, 3001, 3011, 3101, 3111

Offset: 1

Jean-François Alcover, Mar 08 2011

This sequence is similar to A035123 but excludes integers such as 33 or 99 or 3168, because they don't meet the commutativity criterion reverse(n^2) = (reverse(n))^2.
Compare for instance:
{reverse(3168^2), reverse(3168)^2} -> {42263001, 74183769}
with:
{reverse(3111^2), reverse(3111)^2} -> {1238769, 1238769}
Terms can be matched by pairs:
{{12, 21}, {13, 31}, {102, 201}, {103, 301}, {112, 211}, {113, 311}, {122, 221}, {1002, 2001}, {1003, 3001}, {1011, 1101}, {1012, 2101}, {1013, 3101}, {1021, 1201}, {1022, 2201}, {1031, 1301}, {1102, 2011}, {1103, 3011}, {1112, 2111}, {1113, 3111}, {1121, 1211}, {1122, 2211}, {1202, 2021}, {1212, 2121}, {2012, 2102}, {2022, 2202},...}

			113 belongs to the sequence because sqrt(reverse(113^2)) = 311, which is 113 written backwards, whereas 99 does not: sqrt(reverse(99^2)) = 33.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 300 terms from Vincenzo Librandi)

Cf. A064021. Subsequence of A035123.

r[n_] := FromDigits[Reverse[IntegerDigits[n]]];
Cases[Range[10000], n_ /; Mod[n, 10] != 0 && r[n^2] != n^2 && r[n^2] == r[n]^2 ]

a(n)^2 = A064021(n). - Giovanni Resta, Jun 22 2018

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A035090 Non-palindromic squares which when written backwards remain square (and still have the same number of digits).

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

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

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A106324 Greater of number pair whose squares are reversals of each other, with no leading zeros allowed.

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A140212 Numbers n not a multiple of 10 such that reverse(n^2) = reverse(n)^2, but reverse(n) is different from n.

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