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This sequence is a variant of A256739; both sequences have nice graphical features.

Replacing "SumXOR" by "Sum" in the name leads to the Jordan function J_2 (A007434).

For any sequence f of nonnegative integers with positive indices:

- let x_f be the unique sequence satisfying SumXOR_{d divides n} x_f(d) = f(n) for any n > 0,

- in particular, x_A000027 = A256739 and x_A000290 = a (this sequence),

- also, x_A178910 = A000027 and x_A055895 = A000079,

- see the links section for a gallery of x_f plots for some classic f functions,

- x_f(1) = f(1),

- x_f(p) = f(1) XOR f(p) for any prime p,

- x_A063524 = A008966,

- x_f(n) = SumXOR_{d divides n and n/d is squarefree} f(d) for any n > 0,

- the function x: f -> x_f is a bijection,

- A000004 is the only fixed point of x (i.e. x_f = f if and only if f = A000004),

- for any sequence f, x_{2*f} = 2 * x_f,

- for any sequences g and f, x_{g XOR f} = x_g XOR x_f.

From Antti Karttunen, Dec 29 2017: (Start)

The transform x_f described above could be called "Xor-Moebius transform of f" because of its analogous construction to Möbius transform with A008683 replaced by A008966 and the summation replaced by cumulative XOR.

(End)

Unique sequence satisfying SumXOR_{d divides n} a(d) = sigma(n) for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of Xor-Moebius transform.

Unique sequence satisfying SumXOR_{d divides n} a(d) = A006068(n) for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of the Xor-Moebius transform.