A145800 -id:A145800 - cp's OEIS frontend

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

Leroy Quet, Oct 19 2008

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

			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.

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

Cf. A006995, A145800.

Cf. A154809.

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

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 *)

Extended by Ray Chandler, Oct 26 2008

A056544 Smallest palindrome containing the digits of n as a subsequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 101, 11, 121, 131, 141, 151, 161, 171, 181, 191, 202, 121, 22, 232, 242, 252, 262, 272, 282, 292, 303, 131, 232, 33, 343, 353, 363, 373, 383, 393, 404, 141, 242, 343, 44, 454, 464, 474, 484, 494, 505, 151, 252, 353, 454, 55, 565, 575

Offset: 1

Joseph L. Pe, Feb 10 2002

			The smallest palindrome containing 21 as a subsequence is 121. So a(21) = 121.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A056544

Cf. A002113, A082216, A145800 (binary variant).

Do[k = 1; While[ StringPosition[ ToString[k], ToString[n]] == {} || ToString[k] != StringReverse[ ToString[k]], k++ ]; Print[k], {n, 1, 70} ]

PARI
```
See Links section.
```

a(A002113(n)) = A002113(n). - Chai Wah Wu, Apr 13 2021

Edited and extended by Robert G. Wilson v, Feb 12 2002

A161501 a(n) is the smallest positive integer that is a palindrome when represented in binary, and is made by appending some number (zero or more) of binary digits on the right of n's binary representation.

Original entry on oeis.org

1, 5, 3, 9, 5, 27, 7, 17, 9, 21, 45, 51, 27, 119, 15, 33, 17, 73, 153, 165, 21, 45, 93, 99, 51, 107, 27, 231, 119, 495, 31, 65, 33, 273, 561, 73, 297, 153, 313, 325, 165, 85, 693, 717, 45, 93, 189, 195, 99, 403, 51, 843, 107, 219, 443, 455, 231, 471, 119, 975, 495, 2015

Offset: 1

Leroy Quet, Jun 11 2009

			11 (decimal) in binary is 1011. Appending 01 to the right side of 1011 forms the binary palindrome 101101, which is 45 in decimal. Since two binary digits is the smallest number of digits that need to be appended to form a palindrome, then a(11) = 45. (Note that 45 is not the smallest positive number that when represented in binary is a palindrome and contains 1011 as a substring. That would instead be 11011 {binary} = 27 {decimal}.)

Rémy Sigrist, Table of n, a(n) for n = 1..8192

Cf. A006995, A082216 (decimal variant), A145800, A161502.

a(n, base=2) = { my (b=digits(n, base)); if (b==Vecrev(b), return (n), my (t=[]); for (k=1, #b, t=concat(b[k],t); my (bt=concat(b,t)); if (bt==Vecrev(bt), return (fromdigits(bt, base))))) } \\ Rémy Sigrist, Mar 22 2020

def A161501(n):
    s = bin(n)[2:]
    if s == s[::-1]:
        return n
    for i in range(1,len(s)):
        if s[i:] == s[-1:i-1:-1]:
            return int(s+s[i-1::-1],2) # Chai Wah Wu, Aug 27 2021

More terms from Sean A. Irvine, Sep 27 2009

A161502 a(n) is the smallest number of binary digits that when appended to the right side of the binary representation of n, forms a binary palindrome.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 2, 2, 1, 3, 0, 1, 0, 2, 3, 3, 0, 1, 2, 2, 1, 2, 0, 3, 2, 4, 0, 1, 0, 3, 4, 1, 3, 2, 3, 3, 2, 1, 4, 4, 0, 1, 2, 2, 1, 3, 0, 4, 1, 2, 3, 3, 2, 3, 1, 4, 3, 5, 0, 1, 0, 4, 5, 2, 4, 3, 4, 4, 0, 2, 5, 1, 4, 2, 3, 3, 2, 1, 5, 5, 0, 3, 4, 4, 3, 4, 2, 5, 0, 1, 2, 2, 1, 4, 0, 2, 4, 3, 4, 4, 3

Offset: 1

Leroy Quet, Jun 11 2009

			11 (decimal) in binary is 1011. Appending 01 to the right side of 1011 forms the binary palindrome 101101, which is 45 in decimal. Since two binary digits is the smallest number of digits that need to be appended on the right side of binary n to form a palindrome, then a(11) = 2. (Note that 45 is not the smallest positive number that when represented in binary is a palindrome and contains 1011 as a substring. That would instead be 11011 {binary} = 27 {decimal}.)

Cf. A145800, A161501.

def A161502(n):
    s = bin(n)[2:]
    if s == s[::-1]:
        return 0
    for i in range(1,len(s)):
        if s[i:] == s[-1:i-1:-1]:
            return i # Chai Wah Wu, Aug 27 2021

More terms from Sean A. Irvine, Sep 27 2009

cp's OEIS Frontend

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

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A056544 Smallest palindrome containing the digits of n as a subsequence.

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A161501 a(n) is the smallest positive integer that is a palindrome when represented in binary, and is made by appending some number (zero or more) of binary digits on the right of n's binary representation.

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A161502 a(n) is the smallest number of binary digits that when appended to the right side of the binary representation of n, forms a binary palindrome.

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Python

Extensions