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

0,4

Comments

This sequence shares graphical features with A286327.

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

See A327186 for other variants.

Cf. A286327.

Sequence in context: A049901 A117715 A330962 * A160457 A107087 A279955

Adjacent sequences: A327191 A327192 A327193 * A327195 A327196 A327197

Keyword

nonn,look,base

Author

Rémy Sigrist, Aug 25 2019

Status

approved