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
Examples
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.
References
- 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).
Links
- 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
Crossrefs
Programs
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Haskell
a002778 n = a002778_list !! (n-1) a002778_list = filter ((== 1) . a136522 . (^ 2)) [0..] -- Reinhard Zumkeller, Oct 11 2011
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Magma
[n: n in [0..2*10^6] | Intseq(n^2) eq Reverse(Intseq(n^2))]; // Vincenzo Librandi, Apr 07 2015
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Mathematica
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 *)
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PARI
is_A002778(n)=is_A002113(n^2) \\ M. F. Hasler, Jun 08 2014
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Python
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
Extensions
More terms from Patrick De Geest
Comments