A003166 Numbers whose square in base 2 is a palindrome.

0, 1, 3, 4523, 11991, 18197, 141683, 1092489, 3168099, 6435309, 12489657, 17906499, 68301841, 295742437, 390117873, 542959199, 4770504939, 17360493407, 73798050723, 101657343993, 107137400475, 202491428745, 1615452642807, 4902182461643, 9274278357017, 12863364360297

Offset: 1

Author

N. J. A. Sloane, R. H. Hardin

Keywords

• base
• nonn
• hard
• nice

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

Examples

			3^2 = 9 = 1001_2, a palindrome.
4523^2 = 20457529 = 1001110000010100000111001_2.
		

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]

Crossrefs

Cf. A002778 (base 10 analog), A029983 (the actual squares). In binary: A262595, A262596.

Cf. A006995.

Programs

Mathematica

Do[c = RealDigits[n^2, 2][[1]]; If[c == Reverse[c], Print[n]], {n, 0, 10^9}]

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))
A003166_list = list(islice(A003166_gen(),10)) # Chai Wah Wu, Jun 23 2022

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]