A256739 - OEIS

%I #21 Oct 18 2019 11:28:09

%S 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,

%T 20,18,20,28,24,30,48,40,48,32,60,36,54,40,48,40,48,42,60,40,58,46,48,

%U 54,36,32,40,52,54,56,40,40,36,58,48,60,34,32,96,72,120,66

%N 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.

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

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

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

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

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

%C The graph of this sequence is quite remarkable. - _N. J. A. Sloane_, Apr 09 2015

%C 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

%H Paul Tek, <a href="/A256739/b256739.txt">Table of n, a(n) for n = 1..16383</a>

%H Paul Tek, <a href="/A256739/a256739.gp.txt">PARI program for this sequence</a>

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

%F From _Antti Karttunen_, Dec 29 2017: (Start)

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

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

%F (End)

%t a = Table[0, {16383}];

%t Do[pa = n; Do[pa = BitXor[pa, a[[d]]], {d, Divisors[n]}]; a[[n]] = pa, {n, Length[a]}];

%t a (* _Jean-François Alcover_, Oct 18 2019, after _Paul Tek_ *)

%o (PARI) See Links section.

%o 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.

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

%K nonn,base,look

%O 1,2

%A _Paul Tek_, Apr 09 2015