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

Author

Paul Tek, Apr 09 2015

Keywords

• nonn
• base
• look

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

Crossrefs

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

Programs

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 *)

PARI

\\ See Links section.

PARI

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.

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)