A256739 Unique sequence satisfying SumXOR_{d divides n} a(d) = n for any n>0, where SumXOR is the analog of summation under the binary XOR operation.

1, 3, 2, 6, 4, 6, 6, 12, 10, 12, 10, 12, 12, 10, 8, 24, 16, 30, 18, 24, 16, 30, 22, 24, 28, 20, 18, 20, 28, 24, 30, 48, 40, 48, 32, 60, 36, 54, 40, 48, 40, 48, 42, 60, 40, 58, 46, 48, 54, 36, 32, 40, 52, 54, 56, 40, 40, 36, 58, 48, 60, 34, 32, 96, 72, 120, 66

Offset

1,2

Comments

Replacing "SumXOR" by "Sum" in the name leads to the Euler totient function (A000010).

Replacing "SumXOR" by "Product" in the name leads to the exponential of Mangoldt function (A014963).

a(p) = p-1 for any prime p>2.

a(2^k) = 2^k+2^(k-1) for any k>0.

A070939(a(n)) = A070939(n) for any n>0.

The graph of this sequence is quite remarkable. - N. J. A. Sloane, Apr 09 2015

Xor-Moebius transform of natural numbers, A000027. See A295901 for a list of some of the properties of this transform. - Antti Karttunen, Dec 29 2017

Links

Paul Tek, Table of n, a(n) for n = 1..16383

Paul Tek, PARI program for this sequence

Formula

a(n) = n XOR ( SumXOR_{d divides n and d < n} a(d) ) for any n>0.

From Antti Karttunen, Dec 29 2017: (Start)

a(n) = SumXOR_{d|n} A296206(d).

a(n) = n XOR A296207(n), where XOR is bitwise exclusive or, A003987.

(End)

Mathematica

a = Table[0, {16383}];
Do[pa = n; Do[pa = BitXor[pa, a[[d]]], {d, Divisors[n]}]; a[[n]] = pa, {n, Length[a]}];
a (* Jean-François Alcover, Oct 18 2019, after Paul Tek *)

Prog

(PARI) See Links section.
A256739(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, d))); (v); } \\ Antti Karttunen, Dec 29 2017, after code in A295901.

Crossrefs

Cf. A000010, A003987, A014963, A295901, A296206, A296207, A297107 (fixed points).

Sequence in context: A191733 A191444 A258241 * A267104 A093050 A289194

Adjacent sequences: A256736 A256737 A256738 * A256740 A256741 A256742

Keyword

nonn,base,look

Author

Paul Tek, Apr 09 2015

Status

approved