A327189 For any n >= 0: consider the different ways to split the binary representation of n into two (possibly empty) parts, say with value x and y; a(n) is the least possible value of x + y.
0, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 10, 11, 12, 3, 4, 5, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 5
Offset: 0
Examples
For n=42: - the binary representation of 42 is "101010", - there are 7 ways to split it: - "" and "101010": x=0 and y=42: 0 + 42 = 42, - "1" and "01010": x=1 and y=10: 1 + 10 = 11, - "10" and "1010": x=2 and y=10: 2 + 10 = 12, - "101" and "010": x=5 and y=2: 5 + 2 = 7, - "1010" and "10": x=10 and y=2: 10 + 2 = 12, - "10101" and "0": x=21 and y=0: 21 + 0 = 21, - "101010" and "": x=42 and y=0: 42 + 0 = 42, - hence a(42) = 7.
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..8192
Crossrefs
See A327186 for other variants.
Programs
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PARI
a(n) = my (v=oo, b=binary(n)); for (w=0, #b, v=min(v, (fromdigits(b[1..w],2) + fromdigits(b[w+1..#b],2)))); v
Formula
a(n) = 1 iff n is a power of 2.
a(n) = 2 iff n = 2^k + 1 for some k > 0.