A029993 -id:A029993 - cp's OEIS frontend

A002779 Palindromic squares.

0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, 69696, 94249, 698896, 1002001, 1234321, 4008004, 5221225, 6948496, 100020001, 102030201, 104060401, 121242121, 123454321, 125686521, 400080004, 404090404, 522808225

Offset: 1

N. J. A. Sloane

These are numbers that are both squares (see A000290) and palindromes (see A002113).

			676 is included because it is both a perfect square and a palindrome.

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).

Hans Havermann (via Feng Yuan), T. D. Noe (from P. De Geest) [to 485], Table of n, a(n) for n = 1..1940
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012-2014. - 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.
Patrick De Geest, Palindromic Squares in bases 2 to 17
W. R. Marshall, Palindromic Squares
Phakhinkon Phunphayap, Prapanpong Pongsriiam, Reciprocal sum of palindromes, arXiv:1803.00161 [math.CA], 2018.
G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
G. 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. A000290, A002113, A002778, A010052, A027829 (subsets), A028817, A029734.
Cf. A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993.
Cf. A029995, A029997, A029999, A030074, A030075, A057136, A136522.

a002779 n = a002778_list !! (n-1)
a002779_list = filter ((== 1) . a136522) a000290_list
-- Reinhard Zumkeller, Oct 11 2011

[k^2:k in [0..100000]| Intseq(k^2) eq Reverse(Intseq(k^2)) ]; // Marius A. Burtea, Oct 15 2019

palindromicNumberQ = ((# // IntegerDigits // Reverse // FromDigits) == #) &; Select[Table[n^2, {n, 0, 9999}],  palindromicNumberQ] (* Herman Beeksma, Jul 14 2005 *)
pb10Q[n_] := Module[{idn10 = IntegerDigits[n, 10]}, idn10 == Reverse[idn10]]; Select[Range[0, 19999]^2, pb10Q] (* Vincenzo Librandi, Jul 24 2014 *)
Select[Range[0, 22999]^2, PalindromeQ] (* Requires Mathematica version 10 or later. - Harvey P. Dale, May 01 2017 *)

is(n)=my(d=digits(n)); d==Vecrev(d) && issquare(n) \\ Charles R Greathouse IV, Feb 06 2017

A002779_list = [int(s) for s in (str(m**2) for m in range(10**5)) if s == s[::-1]] # Chai Wah Wu, Aug 26 2021

def isPalindromic(n: BigInt): Boolean = n.toString == n.toString.reverse
  val squares = ((1: BigInt) to (1000000: BigInt)).map(n => n * n)
  squares.filter(isPalindromic()) // _Alonso del Arte, Oct 07 2019

From Reinhard Zumkeller, Oct 11 2011: (Start)
a(n) = A002778(n)^2.
A136522(A000290(a(n))) = 1.
A010052(a(n)) * A136522(a(n)) = 1. (End)

A029734 Palindromic squares in base 16.

Original entry on oeis.org

0, 1, 4, 9, 289, 1156, 66049, 74529, 83521, 93025, 101761, 264196, 280900, 298116, 597529, 1896129, 16785409, 19088161, 21538881, 67141636, 68128516, 71673156, 76282756, 81830116, 2665553641, 4295098369, 4328718849

Offset: 1

Patrick De Geest

Chai Wah Wu, Table of n, a(n) for n = 1..377 (terms 1..75 from Vincenzo Librandi)
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029997, A029999, A030074, A030075.

pb16Q[n_]:=Module[{idn16=IntegerDigits[n,16]}, idn16==Reverse[idn16]]; Select[Range[0, 200000]^2, pb16Q] (* Vincenzo Librandi, Jul 24 2014 *)

A029738 Palindromic squares in base 12.

Original entry on oeis.org

0, 1, 4, 9, 169, 676, 21025, 24649, 28561, 32041, 32761, 84100, 85264, 91204, 373321, 2989441, 3045025, 3179089, 3553225, 4165681, 11957764, 13060996, 14409616, 430023169, 436016161, 442050625, 448126561, 505215529

Offset: 1

Patrick De Geest

Vincenzo Librandi, Table of n, a(n) for n = 1..58
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029734, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029997, A029999, A030074, A030075.

pb12Q[n_]:=Module[{idn12=IntegerDigits[n, 12]}, idn12==Reverse[idn12]]; Select[Range[0, 20000]^2, pb12Q] (* Vincenzo Librandi, Jul 24 2014 *)

A029806 n in base 8 is a palindromic square.

Original entry on oeis.org

0, 1, 4, 9, 36, 81, 121, 729, 4225, 5329, 6241, 6561, 6889, 38025, 47961, 56169, 133956, 263169, 294849, 342225, 485809, 1196836, 2368521, 3080025, 3515625, 8468100, 16785409, 17313921, 17850625, 20043529, 21316689, 21911761

Offset: 1

Patrick De Geest

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

Cf. A002779, A029734, A029738, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029997, A029999, A030074, A030075.

pb8Q[n_]:=Module[{idn8=IntegerDigits[n, 8]}, idn8==Reverse[idn8]]; Select[Range[0, 20000]^2, pb8Q] (* Vincenzo Librandi, Jul 24 2014 *)

A029991 Squares which are palindromes in base 6.

Original entry on oeis.org

0, 1, 4, 49, 1369, 1849, 5776, 8281, 47089, 67081, 1132096, 1682209, 1776889, 2307361, 2418025, 7311616, 11336689, 60481729, 64464841, 82428241, 389944009, 2176875649, 2197078129, 2302944121, 2323722025, 2963604721

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

Squares which are palindromes in base b: A029983 (b=2), A029985 (b=3), A029987 (b=4), A029989 (b=5), this sequence (b=6), A029993 (b=7), A029806 (b=8), A029995 (b=9), A002779 (b=10), A029997 (b=11), A029738 (b=12), A029999 (b=13), A030074 (b=14), A030075 (b=15), A029734 (b=16).

Cf. A029990.

a(n) = A029990(n)^2. - Seiichi Manyama, Oct 16 2021

A029995 Squares which are palindromes in base 9.

Original entry on oeis.org

0, 1, 4, 100, 400, 6724, 8281, 10000, 26896, 532900, 672400, 2131600, 43059844, 44129449, 45212176, 53290000, 54479161, 55681444, 172239376, 186104164, 186595600, 203946961, 2921402500, 3486902500, 3583219600

Offset: 1

Patrick De Geest

Vincenzo Librandi, Table of n, a(n) for n = 1..52
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029734, A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029997, A029999, A030074, A030075.

pal9Q[n_]:=Module[{idn9=IntegerDigits[n,9]},idn9==Reverse[idn9]]; Select[ Range[0,60000]^2,pal9Q] (* Harvey P. Dale, Nov 12 2011 *)

A029997 Squares which are palindromes in base 11.

Original entry on oeis.org

0, 1, 4, 9, 36, 144, 576, 676, 5184, 7056, 14884, 17689, 20736, 59536, 65025, 77841, 145924, 535824, 1774224, 2143296, 2547216, 5827396, 7096896, 7817616, 9375844, 20034576, 63872064, 214388164, 217946169, 221533456, 255488256, 259371025

Offset: 1

Patrick De Geest

Vincenzo Librandi and Chai Wah Wu, Table of n, a(n) for n = 1..129 First 60 terms from Vincenzo Librandi.
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029734, A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029999, A030074, A030075.

pb11Q[n_]:=Module[{idn11=IntegerDigits[n,11]},idn11==Reverse[idn11]]; Select[Range[0,17000]^2,pb11Q] (* Harvey P. Dale, Jul 23 2014 *)

from gmpy2 import digits
A029997_list = [n for n in (x**2 for x in range(10**7)) if digits(n,11) == digits(n,11)[::-1]]
# Chai Wah Wu, Dec 01 2014

A029999 Squares which are palindromes in base 13.

Original entry on oeis.org

0, 1, 4, 9, 196, 784, 28900, 33489, 38416, 43681, 94864, 115600, 124609, 133956, 4831204, 5664400, 6563844, 8398404, 16208676, 17994564, 19324816, 20958084, 50098084, 58706244, 815787844, 825470361, 835210000, 845006761, 946915984

Offset: 1

Patrick De Geest

Vincenzo Librandi, Table of n, a(n) for n = 1..61
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029734, A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029997, A030074, A030075.

pb13Q[n_]:=Module[{idn13=IntegerDigits[n, 13]}, idn13==Reverse[idn13]]; Select[Range[0, 20000]^2, pb13Q] (* Vincenzo Librandi, Jul 24 2014 *)

A030074 Squares which are palindromes in base 14.

Original entry on oeis.org

0, 1, 4, 9, 225, 576, 900, 2209, 27225, 38809, 44521, 50625, 57121, 155236, 166464, 178084, 4796100, 5978025, 7535025, 8732025, 10017225, 30140100, 32490000, 73359225, 1475865889, 1490963769, 1506138481, 1521390025

Offset: 1

Patrick De Geest

Vincenzo Librandi, Table of n, a(n) for n = 1..57
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029734, A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029997, A029999, A030075.

pb14Q[n_]:=Module[{idn14=IntegerDigits[n, 14]}, idn14==Reverse[idn14]]; Select[Range[0, 20000]^2, pb14Q] (* Vincenzo Librandi, Jul 24 2014 *)

A030075 Squares which are palindromes in base 15.

Original entry on oeis.org

0, 1, 4, 9, 16, 64, 144, 256, 361, 1024, 1521, 4096, 5776, 16384, 20736, 51076, 58081, 65536, 73441, 96721, 204304, 218089, 228484, 232324, 331776, 511225, 817216, 929296, 1048576, 3055504, 3268864, 3489424, 5308416, 7033104

Offset: 1

Patrick De Geest

			8^2 = 64, which in base 15 is 44, and that's palindromic, so 64 is in the sequence.
9^2 = 81, which in base 15 is 56. Since that's not palindromic, 81 is not in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..106
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A002779, A029734, A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029997, A029999, A030074.

N:= 10^10: # to get all entries <= N
count:= 0:
for x from 0 to floor(sqrt(N)) do
    y:= x^2;
    L:= convert(y,base,15);
  if ListTools[Reverse](L) = L then
     count:= count+1;
     A[count]:= y;
   fi
od:
seq(A[i],i=1..count); # Robert Israel, Jul 24 2014

palQ[n_, b_:10] := Module[{idn = IntegerDigits[n, b]}, idn == Reverse[idn]]; Select[Range[0, 2700]^2, palQ[#, 15] &]  (* Harvey P. Dale, Apr 23 2011 *)

isok(n) = my(d=digits(n,15)); issquare(n) && (d == Vecrev(d)); \\ Michel Marcus, Oct 21 2016

cp's OEIS Frontend

A002779 Palindromic squares.

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A029734 Palindromic squares in base 16.

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A029738 Palindromic squares in base 12.

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A029806 n in base 8 is a palindromic square.

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A029991 Squares which are palindromes in base 6.

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A029995 Squares which are palindromes in base 9.

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A029997 Squares which are palindromes in base 11.

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A029999 Squares which are palindromes in base 13.

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A030074 Squares which are palindromes in base 14.

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