A295901 Unique sequence satisfying SumXOR_{d divides n} a(d) = n^2 for any n > 0, where SumXOR is the analog of summation under the binary XOR operation.

1, 5, 8, 20, 24, 40, 48, 80, 88, 120, 120, 160, 168, 240, 240, 320, 288, 312, 360, 480, 384, 408, 528, 640, 616, 520, 648, 960, 840, 816, 960, 1280, 1072, 1440, 1248, 1248, 1368, 1224, 1360, 1920, 1680, 1920, 1848, 1632, 1872, 2640, 2208, 2560, 2384, 3016

Offset: 1

Author

Rémy Sigrist, Nov 29 2017

Keywords

• nonn
• base

Comments

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)

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..16383
• Rémy Sigrist, Colored scatterplot of the first 2^17-1 terms (where the color is function of A087207(n) % 8)
• Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000010 (Euler totient function)
• Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000203 (sigma)
• Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A001157 (sigma_2)
• Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000040 (prime numbers)
• Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000720 (PrimePi)
• Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A006370 (Collatz map)
• Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A005132 (Recamán's sequence)

Crossrefs

Cf. A000004, A000010, A000027, A000040, A000079, A000290, A000720, A001157, A003987, A005132, A006370, A007434, A008966, A055895, A063524, A087207, A178910.

Same transform applied to other sequences: A256739 (A000027), A296206 (A256739), A296207 (A227320), A296203 (A000203), A296208 (A005187), A297110 (A006068), A297108 (A248663), A297106 (A156552).

Programs

PARI

a(n{, f=k->k^2}) = my (v=0); fordiv(n,d,if (issquarefree(n/d), v=bitxor(v,f(d)))); return (v)

Formula

a(n) = SumXOR_{d divides n and n/d is squarefree} d^2.