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A002778
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Numbers whose square is a palindrome.
(Formerly M0907 N0342)
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72
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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. - 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
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
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EXAMPLE
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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.
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MATHEMATICA
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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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PROG
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(Haskell)
a002778 n = a002778_list !! (n-1)
a002778_list = filter ((== 1) . a136522 . (^ 2)) [0..]
-- Reinhard Zumkeller, Oct 11 2011
(PARI) is_A002778(n)=is_A002113(n^2) \\ M. F. Hasler, Jun 08 2014
(MAGMA) [n: n in [0..2*10^6] | Intseq(n^2) eq Reverse(Intseq(n^2))]; // Vincenzo Librandi, Apr 07 2015
(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
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CROSSREFS
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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.
Sequence in context: A295958 A049083 A305719 * A028816 A316187 A215952
Adjacent sequences: A002775 A002776 A002777 * A002779 A002780 A002781
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KEYWORD
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base,nonn,nice,easy,changed
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Patrick De Geest
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STATUS
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approved
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