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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.
3
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
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
KEYWORD
nonn,look,base
AUTHOR
Rémy Sigrist, Aug 25 2019
STATUS
approved