A327194 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^2 + y^2.

0, 1, 1, 2, 1, 2, 5, 10, 1, 2, 5, 10, 9, 10, 13, 18, 1, 2, 5, 10, 17, 26, 29, 34, 9, 10, 13, 18, 25, 34, 45, 58, 1, 2, 5, 10, 17, 26, 37, 50, 25, 26, 29, 34, 41, 50, 61, 74, 9, 10, 13, 18, 25, 34, 45, 58, 49, 50, 53, 58, 65, 74, 85, 98, 1, 2, 5, 10, 17, 26, 37

Offset: 0

Author

Rémy Sigrist, Aug 25 2019

Keywords

• nonn
• look
• base

Comments

This sequence shares graphical features with A286327.

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: 0^2 + 42^2 = 1764,
   - "1" and "01010": x=1 and y=10: 1^2 + 10^2 = 101,
   - "10" and "1010": x=2 and y=10: 2^2 + 10^2 = 104,
   - "101" and "010": x=5 and y=2: 5^2 + 2^2 = 29,
   - "1010" and "10": x=10 and y=2: 10^2 + 2^2 = 104,
   - "10101" and "0": x=21 and y=0: 21^2 + 0^2 = 441,
   - "101010" and "": x=42 and y=0: 42^2 + 0^2 = 1764,
- hence a(42) = 29.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192

Crossrefs

See A327186 for other variants.

Cf. A286327.

Programs

Mathematica

Table[Min[Total[#^2]&/@Table[FromDigits[#,2]&/@TakeDrop[IntegerDigits[n,2],d],{d,0,IntegerLength[n,2]}]],{n,0,80}] (* Harvey P. Dale, Mar 03 2023 *)

PARI

a(n) = my (v=oo, b=binary(n)); for (w=0, #b, v=min(v, fromdigits(b[1..w],2)^2 + fromdigits(b[w+1..#b],2)^2)); v

Formula

a(n) = 1 iff n is a power of 2.