cp's OEIS Frontend

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.

%I A295901 #26 Dec 30 2017 10:01:38
%S A295901 1,5,8,20,24,40,48,80,88,120,120,160,168,240,240,320,288,312,360,480,
%T A295901 384,408,528,640,616,520,648,960,840,816,960,1280,1072,1440,1248,1248,
%U A295901 1368,1224,1360,1920,1680,1920,1848,1632,1872,2640,2208,2560,2384,3016
%N 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.
%C A295901 This sequence is a variant of A256739; both sequences have nice graphical features.
%C A295901 Replacing "SumXOR" by "Sum" in the name leads to the Jordan function J_2 (A007434).
%C A295901 For any sequence f of nonnegative integers with positive indices:
%C A295901 - let x_f be the unique sequence satisfying SumXOR_{d divides n} x_f(d) = f(n) for any n > 0,
%C A295901 - in particular, x_A000027 = A256739 and x_A000290 = a (this sequence),
%C A295901 - also, x_A178910 = A000027 and x_A055895 = A000079,
%C A295901 - see the links section for a gallery of x_f plots for some classic f functions,
%C A295901 - x_f(1) = f(1),
%C A295901 - x_f(p) = f(1) XOR f(p) for any prime p,
%C A295901 - x_A063524 = A008966,
%C A295901 - x_f(n) = SumXOR_{d divides n and n/d is squarefree} f(d) for any n > 0,
%C A295901 - the function x: f -> x_f is a bijection,
%C A295901 - A000004 is the only fixed point of x (i.e. x_f = f if and only if f = A000004),
%C A295901 - for any sequence f, x_{2*f} = 2 * x_f,
%C A295901 - for any sequences g and f, x_{g XOR f} = x_g XOR x_f.
%C A295901 From _Antti Karttunen_, Dec 29 2017: (Start)
%C A295901 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.
%C A295901 (End)
%H A295901 Rémy Sigrist, <a href="/A295901/b295901.txt">Table of n, a(n) for n = 1..16383</a>
%H A295901 Rémy Sigrist, <a href="/A295901/a295901_7.png">Colored scatterplot of the first 2^17-1 terms</a> (where the color is function of A087207(n) % 8)
%H A295901 Rémy Sigrist, <a href="/A295901/a295901.png">Scatterplot of the first 2^16 terms of x_A000010 (Euler totient function)</a>
%H A295901 Rémy Sigrist, <a href="/A295901/a295901_1.png">Scatterplot of the first 2^16 terms of x_A000203 (sigma)</a>
%H A295901 Rémy Sigrist, <a href="/A295901/a295901_2.png">Scatterplot of the first 2^16 terms of x_A001157 (sigma_2)</a>
%H A295901 Rémy Sigrist, <a href="/A295901/a295901_3.png">Scatterplot of the first 2^16 terms of x_A000040 (prime numbers)</a>
%H A295901 Rémy Sigrist, <a href="/A295901/a295901_4.png">Scatterplot of the first 2^16 terms of x_A000720 (PrimePi)</a>
%H A295901 Rémy Sigrist, <a href="/A295901/a295901_5.png">Scatterplot of the first 2^16 terms of x_A006370 (Collatz map)</a>
%H A295901 Rémy Sigrist, <a href="/A295901/a295901_6.png">Scatterplot of the first 2^16 terms of x_A005132 (Recamán's sequence)</a>
%F A295901 a(n) = SumXOR_{d divides n and n/d is squarefree} d^2.
%o A295901 (PARI) a(n{, f=k->k^2}) = my (v=0); fordiv(n,d,if (issquarefree(n/d), v=bitxor(v,f(d)))); return (v)
%Y A295901 Cf. A000004, A000010, A000027, A000040, A000079, A000290, A000720, A001157, A003987, A005132, A006370, A007434, A008966, A055895, A063524, A087207, A178910.
%Y A295901 Same transform applied to other sequences: A256739 (A000027), A296206 (A256739), A296207 (A227320), A296203 (A000203), A296208 (A005187), A297110 (A006068), A297108 (A248663), A297106 (A156552).
%K A295901 nonn,base
%O A295901 1,2
%A A295901 _Rémy Sigrist_, Nov 29 2017