OFFSET
1,3
COMMENTS
Numbers k such that k^2 is in A006995.
The only palindromes in this sequence are 0, 1, and 3. See AMM problem 11922. - Max Alekseyev, Oct 22 2022
REFERENCES
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Don Knuth, Table of n, a(n) for n = 1..50 [This table extends earlier work of Gus Simmons, Jon Schoenfield, Don Knuth, and Michael Coriand]
M. A. Alekseyev, Problem 11922. American Mathematical Monthly 123:7 (2016), 722.
Patrick De Geest, Palindromic Squares in bases 2 to 17
Carlos Rivera, Problem 89. Palindromic binary expression of primes squared, The Prime Puzzles & Problems Connection.
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]
EXAMPLE
3^2 = 9 = 1001_2, a palindrome.
4523^2 = 20457529 = 1001110000010100000111001_2.
MATHEMATICA
Do[c = RealDigits[n^2, 2][[1]]; If[c == Reverse[c], Print[n]], {n, 0, 10^9}]
PROG
(PARI) is(n)=my(b=binary(n^2)); b==Vecrev(b) \\ Charles R Greathouse IV, Feb 07 2017
(Python)
from itertools import count, islice
def A003166_gen(): # generator of terms
return filter(lambda k: (s:=bin(k**2)[2:])[:(t:=(len(s)+1)//2)]==s[:-t-1:-1], count(0))
CROSSREFS
KEYWORD
base,nonn,hard,nice,changed
AUTHOR
EXTENSIONS
a(16) = 4770504939 found by Patrick De Geest, May 15 1999
a(17)-a(31) from Jon E. Schoenfield, May 08 2009
a(32) = 285000288617375,
a(33) = 301429589329949,
a(34) = 1178448744881657 from Don Knuth, Jan 28 2013 [who doublechecked the previous results and searched up to 2^104]
STATUS
approved