A027829 Palindromic squares with an even number of digits.

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

Offset

1,1

References

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]

Links

M. F. Hasler, Table of n, a(n) for n = 1..14

K. S. Brown, On General Palindromic Numbers

Patrick De Geest, Palindromic Squares

Patrick De Geest, Subsets of Palindromic Squares

Michael Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132. [Annotated scanned copy]

Formula

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

Example

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.

Mathematica

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

Prog

(PARI) 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
(Scala) 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
(Python)
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

Crossrefs

Cf. A002113, A002778, A002779, A016113.

Sequence in context: A319917 A205608 A205439 * A258129 A204496 A332850

Adjacent sequences: A027826 A027827 A027828 * A027830 A027831 A027832

Keyword

nonn,base

Author

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

Extensions

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

Status

approved