A298481 - OEIS

%I #16 Mar 10 2018 18:24:19

%S 1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%T 1,2,2,2,1,1,2,3,1,2,1,1,1,1,1,2,1,2,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%U 2,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1

%N a(n) is the number of ways to partition the binary representation of n into the minimal number of palindromic parts.

%C A minimal palindromic partition is a partition of the string into palindromes with the fewest parts.

%C A298475(n) gives the size of the minimal partition of the binary representation of n.

%C Records occur at 1, 10, 42, 170, 682, 1357, 5428, 5453, 21812, 21837, 45746, ....

%H Peter Kagey, <a href="/A298481/b298481.txt">Table of n, a(n) for n = 1..10000</a>

%e The a(n) minimal palindromic partitions for five integers:

%e   n | a(n) | A298475(n) | binary   | partitions

%e ----+------+------------+----------+--------------------------------------

%e 2   | 1    | 2          | 10       | 1'0

%e 5   | 1    | 1          | 101      | 101

%e 10  | 2    | 2          | 1010     | 101'0 or 1'010

%e 37  | 2    | 3          | 100101   | 1001'0'1 or 1'00'101

%e 149 | 3    | 3          | 10010101 | 1001'010'1, 1'00'10101, or 1001'0'101

%t {1, 1}~Join~Array[Function[w, Length@ MinimalBy[#, Length] &@ Select[#, And[AllTrue[#, PalindromeQ], Union@ Map[Length, #] != {1}] &] &@ Union@ Map[Select[SplitBy[#, IntegerQ], IntegerQ@ First@ # &] &, Map[Insert[w, ".", #] &, Map[{#} &, Rest@ Subsets@ Range@ Length@ w, {2}]]]]@ IntegerDigits[#, 2] &, 103, 3] (* _Michael De Vlieger_, Jan 23 2018 *)

%Y Cf. A298475.

%K nonn,base

%O 1,10

%A _Peter Kagey_, Jan 19 2018