A003166 -id:A003166 - cp's OEIS frontend

A002778 Numbers whose square is a palindrome.

0, 1, 2, 3, 11, 22, 26, 101, 111, 121, 202, 212, 264, 307, 836, 1001, 1111, 2002, 2285, 2636, 10001, 10101, 10201, 11011, 11111, 11211, 20002, 20102, 22865, 24846, 30693, 100001, 101101, 110011, 111111, 200002, 798644, 1000001, 1001001

Offset: 1

N. J. A. Sloane

A002779(n) = a(n)^2; A136522(A000290(a(n))) = 1. - Reinhard Zumkeller, Oct 11 2011

See A016113 for the subset of numbers whose palindromic squares have an even number of digits. - M. F. Hasler, Jun 08 2014

			26^2 = 676, which is a palindrome, so 26 is in the sequence.
27^2 = 729, which is not a palindrome, so 27 is not in the sequence.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Patrick De Geest, Table of n, a(n) for n = 1..8729 (communicated by Max Alekseyev)
Patrick De Geest, Table of square palindromes
Patrick De Geest, Palindromic Squares in bases 2 to 17
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012. - From _N. J. A. Sloane_, Nov 08 2012
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.5.
Michael Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132. [Annotated scanned copy]
William Rex Marshall, Palindromic Squares
Gustavus J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
Gustavus J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Palindromic Number.
Feng Yuan, Palindromic Square Numbers

Cf. A002779, A002113, A016113, A136522, A000290.
See A003166 for binary analog.
For analogs in bases 2,3,4,5,etc. see A003166 onwards, A029984 onwards, and A263607 onwards.

a002778 n = a002778_list !! (n-1)
a002778_list = filter ((== 1) . a136522 . (^ 2)) [0..]
-- Reinhard Zumkeller, Oct 11 2011

[n: n in [0..2*10^6] | Intseq(n^2) eq Reverse(Intseq(n^2))]; // Vincenzo Librandi, Apr 07 2015

palsquareQ[n_] := (n2 = IntegerDigits[n^2]; n2 == Reverse[n2]); A002778 = {}; Do[ If[palsquareQ[n], Print[n]; AppendTo[A002778, n]], {n, 0, 2 * 10^6}]; A002778 (* Jean-François Alcover, Dec 01 2011 *)
Sqrt[#]&/@Select[Range[0, 12 * 10^5]^2, # == IntegerReverse[#] &] (* The program uses the IntegerReverse function from Mathematica version 10. - Harvey P. Dale, Mar 04 2016 *)
Select[Range[0, 1001001], PalindromeQ[#^2] &] (* Michael De Vlieger, Dec 06 2017 *)

is_A002778(n)=is_A002113(n^2) \\ M. F. Hasler, Jun 08 2014

from itertools import count, islice
def A002778_gen(): # generator of terms
    return filter(lambda k: (s:=str(k**2))[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],count(0))
A002778_list = list(islice(A002778_gen(),20)) # Chai Wah Wu, Jun 23 2022

More terms from Patrick De Geest

A029984 Numbers k such that k^2 is palindromic in base 3.

Original entry on oeis.org

0, 1, 2, 4, 10, 11, 20, 22, 28, 34, 56, 82, 89, 113, 154, 164, 244, 262, 488, 524, 730, 755, 802, 862, 1021, 1342, 1358, 1460, 2188, 2242, 2684, 2716, 3046, 4276, 4376, 4484, 6562, 6641, 6778, 8030, 8215, 8350, 8887, 12482, 13124, 14810, 19684

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..400 (first 100 terms from Harvey P. Dale)
Patrick De Geest, Palindromic Squares in bases 2 to 17
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778, A003166, A007089, A029985, A263607, A263608.

pal3Q[n_]:=Module[{idn3=IntegerDigits[n^2,3]},idn3==Reverse[idn3]]; Select[Range[0,20000],pal3Q]  (* Harvey P. Dale, May 22 2012 *)

A029983 Squares which are palindromes in base 2.

Original entry on oeis.org

0, 1, 9, 20457529, 143784081, 331130809, 20074072489, 1193532215121, 10036851273801, 41413201925481, 155991531977649, 320642706437001, 4665141483989281, 87463589042698969, 152191954834044129

Offset: 1

Patrick De Geest

Chai Wah Wu, Table of n, a(n) for n = 1..22
Patrick De Geest, Palindromic Squares in bases 2 to 17
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A003166.

from itertools import count, islice
def A029983_gen(): # generator of terms
    return filter(lambda k: (s:=bin(k)[2:])[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],(k**2 for k in count(0)))
A029983_list = list(islice(A029983_gen(),10)) # Chai Wah Wu, Jun 23 2022

A029989 Squares which are palindromes in base 5.

Original entry on oeis.org

0, 1, 4, 36, 676, 961, 4356, 15876, 24336, 391876, 423801, 571536, 646416, 9771876, 10732176, 14107536, 81974916, 244171876, 248094001, 264908176, 339812356, 351787536, 1061326084, 1167042244, 2181263616, 6103671876

Offset: 1

Patrick De Geest

Chai Wah Wu, Table of n, a(n) for n = 1..147
Patrick De Geest, Palindromic Squares in bases 2 to 17
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778, A003166, A263611, A263612, A029988.

pal5Q[n_]:=Module[{idn5=IntegerDigits[n,5]},idn5==Reverse[idn5]]; Select[ Range[0,80000]^2,pal5Q] (* Harvey P. Dale, May 05 2012 *)

A029985 Squares which are palindromes in base 3.

Original entry on oeis.org

0, 1, 4, 16, 100, 121, 400, 484, 784, 1156, 3136, 6724, 7921, 12769, 23716, 26896, 59536, 68644, 238144, 274576, 532900, 570025, 643204, 743044, 1042441, 1800964, 1844164, 2131600, 4787344, 5026564, 7203856, 7376656, 9278116

Offset: 1

Patrick De Geest

Chai Wah Wu, Table of n, a(n) for n = 1..521
Patrick De Geest, Palindromic Squares in bases 2 to 17
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A029984, A263607, A263608; also A002778, A003166.

b3pQ[n_]:=Module[{idn3=IntegerDigits[n,3]},idn3==Reverse[idn3]]; Select[ Range[0,3200]^2,b3pQ] (* Harvey P. Dale, Aug 07 2011 *)

A029986 Numbers k such that k^2 is palindromic in base 4.

Original entry on oeis.org

0, 1, 5, 17, 21, 65, 71, 83, 257, 273, 281, 317, 1025, 1055, 4097, 4161, 4193, 4401, 5157, 5179, 5221, 16385, 16511, 16865, 17239, 65537, 65793, 65921, 66753, 68695, 69521, 69777, 80739, 82053, 82171, 82309, 82885, 83301, 262145

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A007090, A014192.

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), this sequence (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), A029805 (b=8), A029994 (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), A029998 (b=13), A030072 (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

Select[Range[0,300000],IntegerDigits[#^2,4]==Reverse[ IntegerDigits[ #^2,4]]&] (* Harvey P. Dale, Dec 01 2015 *)

isok(k) = my(d=digits(k^2,4)); d == Vecrev(d); \\ Michel Marcus, Jul 04 2021

A029988 Numbers k such that k^2 is palindromic in base 5.

Original entry on oeis.org

0, 1, 2, 6, 26, 31, 66, 126, 156, 626, 651, 756, 804, 3126, 3276, 3756, 9054, 15626, 15751, 16276, 18434, 18756, 32578, 34162, 46704, 78126, 78876, 81276, 93756, 390626, 391251, 393876, 406276, 468756, 487981, 1166454, 1953126, 1956876

Offset: 1

Patrick De Geest

Patrick De Geest, Palindromic Squares in bases 2 to 17
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778, A003166, A007091, A263611, A263612, A029989.

pal5Q[n_]:=Module[{idn5=IntegerDigits[n^2,5]},idn5==Reverse[idn5]]; Select[ Range[ 0,2*10^6],pal5Q] (* Harvey P. Dale, Feb 02 2023 *)

A029990 Numbers k such that k^2 is palindromic in base 6.

Original entry on oeis.org

0, 1, 2, 7, 37, 43, 76, 91, 217, 259, 1064, 1297, 1333, 1519, 1555, 2704, 3367, 7777, 8029, 9079, 19747, 46657, 46873, 47989, 48205, 54439, 54655, 54695, 83979, 118027, 241304, 279937, 281449, 287749, 326599, 707707, 1679617

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..92
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A007092.

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), this sequence (b=6), A029992 (b=7), A029805 (b=8), A029994 (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), A029998 (b=13), A030072 (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

palindromicQ[n_, b_:10] := TrueQ[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]]; Select[Range[1000], palindromicQ[#^2, 6] &] (* Alonso del Arte, Mar 05 2017 *)

ispal(n,base)=my(d=digits(n,base)); d==Vecrev(d)
is(n)==ispal(n^2,6) \\ Charles R Greathouse IV, Mar 09 2017

A029992 Numbers k such that k^2 is palindromic in base 7.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 11, 20, 32, 40, 50, 57, 64, 80, 160, 200, 344, 400, 500, 550, 557, 730, 1000, 1376, 1432, 1892, 2402, 2451, 2500, 2752, 2801, 3440, 3784, 3902, 5101, 5266, 6880, 8296, 9460, 9608, 9804, 16808, 17200, 19216, 19608, 22693

Offset: 1

Patrick De Geest

			8^2 = 64, which is 121 in base 7, and since that's palindromic, 8 is in the sequence.
9^2 = 81, which is 144 in base 7, but since that's not palindromic, 9 is not in the sequence.

Marius A. Burtea, Table of n, a(n) for n = 1..237
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002440 (squares written in base 7), A007093.

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), this sequence (b=7), A029805 (b=8), A029994 (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), A029998 (b=13), A030072 (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

[k:k in [0..23000]| Seqint(Intseq(k^2,7)) eq Seqint(Reverse(Intseq(k^2,7)))]; // Marius A. Burtea, Jan 22 2020

Select[Range[0, 16806], IntegerDigits[#^2, 7] == Reverse[IntegerDigits[#^2, 7]] &] (* Alonso del Arte, Jan 21 2020 *)

(0 to 16806).filter(n => Integer.toString(n * n, 7) == Integer.toString(n * n, 7).reverse) // Alonso del Arte, Jan 21 2020

A029733 Numbers k such that k^2 is palindromic in base 16.

Original entry on oeis.org

0, 1, 2, 3, 17, 34, 257, 273, 289, 305, 319, 514, 530, 546, 773, 1377, 4097, 4369, 4641, 8194, 8254, 8466, 8734, 9046, 51629, 65537, 65793, 66049, 66305, 69649, 69905, 70161, 70417, 73505, 73761, 74017, 74273, 76879, 92327, 131074

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), A029805 (b=8), A029994 (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), A029998 (b=13), A030072 (b=14), A030073 (b=15), this sequence (b=16), A118651 (b=17).

n2palQ[n_]:=Module[{id=IntegerDigits[n^2,16]},id==Reverse[id]]; Select[ Range[ 0,150000],n2palQ] (* Harvey P. Dale, Mar 31 2018 *)

from itertools import count, islice
def A029733_gen(): # generator of terms
    return filter(lambda k: (s:=hex(k**2)[2:])[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],count(0))
A029733_list = list(islice(A029733_gen(),20)) # Chai Wah Wu, Jun 23 2022

cp's OEIS Frontend

A002778 Numbers whose square is a palindrome.

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A029984 Numbers k such that k^2 is palindromic in base 3.

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A029983 Squares which are palindromes in base 2.

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A029989 Squares which are palindromes in base 5.

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A029985 Squares which are palindromes in base 3.

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A029986 Numbers k such that k^2 is palindromic in base 4.

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A029988 Numbers k such that k^2 is palindromic in base 5.

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A029990 Numbers k such that k^2 is palindromic in base 6.

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A029992 Numbers k such that k^2 is palindromic in base 7.

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