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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.

%I A256739 #24 Jul 25 2025 19:01:57
%S A256739 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 A256739 20,18,20,28,24,30,48,40,48,32,60,36,54,40,48,40,48,42,60,40,58,46,48,
%U A256739 54,36,32,40,52,54,56,40,40,36,58,48,60,34,32,96,72,120,66
%N 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.
%C A256739 Replacing "SumXOR" by "Sum" in the name leads to the Euler totient function (A000010).
%C A256739 Replacing "SumXOR" by "Product" in the name leads to the exponential of Mangoldt function (A014963).
%C A256739 a(p) = p-1 for any prime p>2.
%C A256739 a(2^k) = 2^k+2^(k-1) for any k>0.
%C A256739 A070939(a(n)) = A070939(n) for any n>0.
%C A256739 The graph of this sequence is quite remarkable. - _N. J. A. Sloane_, Apr 09 2015
%C A256739 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 A256739 Paul Tek, <a href="/A256739/b256739.txt">Table of n, a(n) for n = 1..16383</a>
%H A256739 Paul Tek, <a href="/A256739/a256739.gp.txt">PARI program for this sequence</a>
%F A256739 a(n) = n XOR ( SumXOR_{d divides n and d < n} a(d) ) for any n>0.
%F A256739 From _Antti Karttunen_, Dec 29 2017: (Start)
%F A256739 a(n) = SumXOR_{d|n} A296206(d).
%F A256739 a(n) = n XOR A296207(n), where XOR is bitwise exclusive or, A003987.
%F A256739 (End)
%t A256739 a = Table[0, {16383}];
%t A256739 Do[pa = n; Do[pa = BitXor[pa, a[[d]]], {d, Divisors[n]}]; a[[n]] = pa, {n, Length[a]}];
%t A256739 a (* _Jean-François Alcover_, Oct 18 2019, after _Paul Tek_ *)
%o A256739 (PARI) \\ See Links section.
%o A256739 (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.
%Y A256739 Cf. A000010, A003987, A014963, A295901, A296206, A296207, A297107 (fixed points).
%K A256739 nonn,base,look
%O A256739 1,2
%A A256739 _Paul Tek_, Apr 09 2015