A145799 - OEIS

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

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OFFSET

1,3

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

LINKS

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A006995, A145800.