A370427 - OEIS

A370427

a(n) is the least k >= 0 such that n OR k is a binary palindrome (where OR denotes the bitwise OR operator).

%I #13 Feb 21 2024 02:18:11

%S 0,0,1,0,1,0,1,0,1,0,5,4,3,2,1,0,1,0,9,8,1,0,9,8,3,2,1,0,3,2,1,0,1,0,

%T 17,16,9,8,25,24,5,4,21,20,1,0,17,16,3,2,1,0,11,10,9,8,7,6,5,4,3,2,1,

%U 0,1,0,33,32,17,16,49,48,1,0,33,32,17,16,49,48

%N a(n) is the least k >= 0 such that n OR k is a binary palindrome (where OR denotes the bitwise OR operator).

%C The binary expansions of n and a(n) have no common 1's.

%H Paolo Xausa, <a href="/A370427/b370427.txt">Table of n, a(n) for n = 0..16383</a>

%H Rémy Sigrist, <a href="/A370427/a370427.png">Scatterplot of (x, y) such that x OR y is a binary palindrome and x, y < 2^10</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/Pac#palindromes">Index entries for sequences related to palindromes</a>

%F n AND a(n) = 0 (where AND denotes the bitwise AND operator).

%F a(n) = A030101(n) - (n AND A030101(n)).

%F a(n) = A030101(n) - A175297(n) (for any n > 0).

%F a(n) = 0 iff n belongs to A006995.

%e The first terms, alongside the corresponding binary expansions, are:

%e n a(n) bin(n) bin(a(n)) bin(n OR a(n))

%e -- ---- ------ --------- --------------

%e 0 0 0 0 0

%e 1 0 1 0 1

%e 2 1 10 1 11

%e 3 0 11 0 11

%e 4 1 100 1 101

%e 5 0 101 0 101

%e 6 1 110 1 111

%e 7 0 111 0 111

%e 8 1 1000 1 1001

%e 9 0 1001 0 1001

%e 10 5 1010 101 1111

%e 11 4 1011 100 1111

%e 12 3 1100 11 1111

%e 13 2 1101 10 1111

%e 14 1 1110 1 1111

%e 15 0 1111 0 1111

%t A370427[n_] := With[{r = IntegerReverse[n, 2]}, r - BitAnd[n, r]];

%t Array[A370427, 2^7, 0] (* _Paolo Xausa_, Feb 20 2024 *)

%o (PARI) a(n) = my (r = fromdigits(Vecrev(binary(n)), 2)); r - bitand(n, r)

%Y Cf. A006995, A030101, A175297, A344220 (XOR variant).

%K nonn,base,easy,look

%O 0,11

%A _Rémy Sigrist_, Feb 18 2024