A145799 a(n) = the largest integer that is an (odd) palindrome when represented in binary and that occurs in the binary representation of n.

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 5, 3, 5, 7, 15, 1, 17, 9, 9, 5, 21, 5, 7, 3, 9, 5, 27, 7, 7, 15, 31, 1, 33, 17, 17, 9, 9, 9, 9, 5, 9, 21, 21, 5, 45, 7, 15, 3, 17, 9, 51, 5, 21, 27, 27, 7, 9, 7, 27, 15, 15, 31, 63, 1, 65, 33, 33, 17, 17, 17, 17, 9, 73, 9, 9, 9, 9, 9, 15, 5, 17, 9, 9, 21, 85, 21, 21

Offset: 1

Author

Leroy Quet, Oct 19 2008

Keywords

• base
• nonn
• look

Comments

The binary expansion of a(n) is the largest (odd) palindrome that appears as a substring of the binary expansion of n. Nonzero binary palindromes are necessarily odd (see A006995).

For n = 2^k, a(n) = 1 is the largest binary palindrome in the binary representation of n.

a(2^k*A006995(n)) = A006995(n). - Ray Chandler, Oct 26 2008

a(m) = m iff m is a palindrome: a(A006995(n)) = A006995(n), a(A154809(n)) < A154809(n). - Reinhard Zumkeller, Sep 24 2015

Examples

			20 in binary is 10100. The largest binary palindrome included in this binary representation is 101, which is 5 in decimal. So a(20) = 5.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A006995, A145800.

Cf. A154809.

Programs

Haskell

a145799 = maximum . map (foldr (\b v -> 2 * v + b) 0) .
                    filter (\bs -> bs == reverse bs && head bs == 1) .
                    substr . bin where
   substr [] = []
   substr us'@(_:us) = sub us' ++ substr us where
      sub [] = []; sub (v:vs) = [v] : [v : ws | ws <- sub vs ]
   bin 0 = []; bin n = b : bin n' where (n', b) = divMod n 2
-- Reinhard Zumkeller, Sep 24 2015

Mathematica

Block[{nn = 87, s}, s = Reverse@ Select[IntegerDigits[#, 2] & /@ Range[2^Log2@ nn], PalindromeQ]; Table[With[{d = IntegerDigits[n, 2]}, FromDigits[#, 2] &@ SelectFirst[s, SequenceCount[d, #] > 0 &]], {n, nn}]] (* Michael De Vlieger, Sep 23 2017 *)

Extensions

Extended by Ray Chandler, Oct 26 2008