A016113 -id:A016113 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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.

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).

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

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.

a002778 n = a002778_list !! (n-1)
a002778_list = filter ((== 1) . a136522 . (^ 2)) [0..]
-- Reinhard Zumkeller, Oct 11 2011

[n: n in [0..2*10^6] | Intseq(n^2) eq Reverse(Intseq(n^2))]; // Vincenzo Librandi, Apr 07 2015

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 *)

is_A002778(n)=is_A002113(n^2) \\ M. F. Hasler, Jun 08 2014

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

More terms from Patrick De Geest

A027829 Palindromic squares with an even number of digits.

Original entry on oeis.org

698896, 637832238736, 4099923883299904, 6916103777337773016196, 40460195511188111559106404, 4872133543202112023453312784, 9658137819052882509187318569, 46501623417708833880771432610564, 1635977102407987117897042017795361, 163296619873968186681869378916692361

Offset: 1

Keith Devlin, via Boon Leong (boon_leong(AT)hotmail.com)

			836^2 = 698896, which is palindromic, so 698896 is in the sequence.
1001^2 = 1002001, which is palindromic, but it has an odd number of digits, so it's not in the sequence.

Charles Ashbacher, More on palindromic squares, J. Rec. Math. 22, no. 2 (1990), 133-135. [A scan of the first page of this article is included with the last page of the Keith (1990) scan]

M. F. Hasler, Table of n, a(n) for n = 1..14
K. S. Brown, On General Palindromic Numbers
Patrick De Geest, Sporadic palindromic squares of even length
Michael Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132. [Annotated scanned copy]

Cf. A002113, A002778, A002779, A016113.

Select[Range[1000000]^2, PalindromeQ[#] && OddQ[Floor[Log[10, #]]] &] (* Alonso del Arte, Oct 11 2019 *)

is_A027829(n)={issquare(n)&&Vecrev(n=digits(n))==n&&!bittest(#n, 0)} \\ This is faster than first checking for even length if applied to numbers known to have an even number of digits, as should be the case for a systematic search. For this, one should only consider squares, i.e., rather use is_A016113.  - M. F. Hasler, Jun 08 2014

from math import isqrt
from itertools import count, islice
def A027829_gen(): # generator of terms
    return filter(lambda n: (s:=str(n))[:(t:=(len(s)+1)//2)]==s[:-t-1:-1], map(lambda n: n**2, (d for l in count(2,2) for d in range(isqrt(10**(l-1))+1,isqrt(10**l)+1))))
A027829_list = list(islice(A027829_gen(),3)) # Chai Wah Wu, Jun 23 2022

def isPalindromic(n: BigInt): Boolean = n.toString == n.toString.reverse
val squares = ((1: BigInt) to (1000000: BigInt)).map(n => n * n)
squares.filter(n => isPalindromic(n) && n.toString.length % 2 == 0) // Alonso del Arte, Oct 07 2019

a(n) = A016113(n)^2. - M. F. Hasler, Jun 08 2014

Two new terms were recently found by Bennett from UK (communication from Patrick De Geest, Dec 1999 or before)

Edited by M. F. Hasler, Jun 08 2014

A065378 Primes p such that p^2 is a palindromic square.

Original entry on oeis.org

2, 3, 11, 101, 307, 30001253, 100111001, 110111011, 111010111, 111091111, 1011099011101, 1100011100011, 1100101010011, 1101010101011, 100110101011001, 100110990111001, 101000010000101, 101011000110101, 101110000011101

Offset: 1

Patrick De Geest, Nov 03 2001

Record prime base number of a sporadic palindromic square is 13661181333262459.

			E.g. a(6) = 900075181570009 = p^2 with p = 30001253 and prime.

Hans Havermann (via Feng Yuan), Table of n, a(n) for n = 1..61
Patrick De Geest, Palindromic Squares in bases 2 to 17
G. L. Honaker, Jr. and C. Caldwell, Prime Curios!
W. R. Marshall, Palindromic Squares
F. Yuan, Palindromic Square Numbers

Cf. A065379, A002778, A002779, A034822, A028816, A016106, A059744, A016113, A007573.

t={}; Do[p=Prime[n]; If[FromDigits[Reverse[IntegerDigits[p^2]]]==p^2,AppendTo[t,p]],{n,10^7}]; t (* Jayanta Basu, May 11 2013 *)

Corrected by Jayanta Basu, May 11 2013

A065379 Palindromic squares with a prime root.

Original entry on oeis.org

4, 9, 121, 10201, 94249, 900075181570009, 12124434743442121, 12323244744232321, 12341234943214321, 1022321210249420121232201, 1210024420147410244200121, 1210222232227222322220121

Offset: 1

Patrick De Geest, Nov 03 2001

Record sporadic palindromic square with a prime root is 186627875420278656872024578726681.

			a(6) = 900075181570009 = p^2 with p = 30001253, a prime.

Patrick De Geest, Palindromic Squares in bases 2 to 17
G. L. Honaker, Jr. and C. Caldwell, Prime Curios!
William Rex Marshall, Palindromic Squares [Internet Archive Wayback Machine]

Cf. A065378, A002778, A002779, A034822, A028816, A016106, A059744, A016113, A007573.

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A002778 Numbers whose square is a palindrome.

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A027829 Palindromic squares with an even number of digits.

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A065378 Primes p such that p^2 is a palindromic square.

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A065379 Palindromic squares with a prime root.

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