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A002779 Palindromic squares.
(Formerly M3371 N1358)
41
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
These are numbers that are both squares (see A000290) and palindromes (see A002113).
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
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
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.
FORMULA
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)
EXAMPLE
676 is included because it is both a perfect square and a palindrome.
MATHEMATICA
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 *)
PROG
(Haskell)
a002779 n = a002778_list !! (n-1)
a002779_list = filter ((== 1) . a136522) a000290_list
-- Reinhard Zumkeller, Oct 11 2011
(PARI) is(n)=my(d=digits(n)); d==Vecrev(d) && issquare(n) \\ Charles R Greathouse IV, Feb 06 2017
(Scala) 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
(Magma) [k^2:k in [0..100000]| Intseq(k^2) eq Reverse(Intseq(k^2)) ]; // Marius A. Burtea, Oct 15 2019
(Python)
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
CROSSREFS
Sequence in context: A229971 A158642 A131760 * A028817 A319483 A057136
KEYWORD
nonn,base,nice,easy
AUTHOR
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)