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A027829 Palindromic squares with an even number of digits. 4
698896, 637832238736, 4099923883299904, 6916103777337773016196, 40460195511188111559106404, 4872133543202112023453312784, 9658137819052882509187318569, 46501623417708833880771432610564, 1635977102407987117897042017795361, 163296619873968186681869378916692361 (list; graph; refs; listen; history; text; internal format)
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
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]
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
Sequence in context: A319917 A205608 A205439 * A258129 A204496 A332850
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

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Last modified July 14 02:38 EDT 2024. Contains 374291 sequences. (Running on oeis4.)