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A003166
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Numbers whose square in base 2 is a palindrome.
(Formerly M3181)
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24
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0, 1, 3, 4523, 11991, 18197, 141683, 1092489, 3168099, 6435309, 12489657, 17906499, 68301841, 295742437, 390117873, 542959199, 4770504939, 17360493407, 73798050723, 101657343993, 107137400475, 202491428745, 1615452642807
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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REFERENCES
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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).
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LINKS
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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]
Patrick De Geest, Palindromic Squares
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]
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EXAMPLE
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3^2 = 9 = 1001_2, a palindrome.
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MATHEMATICA
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Do[c = RealDigits[n^2, 2][[1]]; If[c == Reverse[c], Print[n]], {n, 0, 10^9}]
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PROG
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(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
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CROSSREFS
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Cf. A002778 (base 10 analog), A029983 (the actual squares). In binary: A262595, A262596.
Sequence in context: A024048 A094319 A229766 * A251603 A168556 A200950
Adjacent sequences: A003163 A003164 A003165 * A003167 A003168 A003169
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KEYWORD
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base,nonn,hard,nice,changed
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AUTHOR
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N. J. A. Sloane, R. H. Hardin
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EXTENSIONS
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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]
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STATUS
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approved
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