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A002779 Palindromic squares.
(Formerly M3371 N1358)
39
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).

a(n) = A002778(n)^2; A136522(A000290(a(n))) = 1. - Reinhard Zumkeller, Oct 11 2011

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.

Feng Yuan, Palindromic Square Numbers

FORMULA

A010052(a(n)) * A136522(a(n)) = 1. - Reinhard Zumkeller, Oct 11 2011

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

Cf. A000290, A002778, A002113, A057136, A136532, A010052, A029734, A029738, A029806, A029983, A029985, A029987, A029989, A029991, A029993, A029995, A029997, A029999, A030074, A030075.

Cf. A028817, A027829 (subsets).

Sequence in context: A229971 A158642 A131760 * A028817 A319483 A057136

Adjacent sequences:  A002776 A002777 A002778 * A002780 A002781 A002782

KEYWORD

nonn,base,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 26 21:14 EDT 2022. Contains 354885 sequences. (Running on oeis4.)