A297106 - OEIS

%I #11 Dec 26 2017 17:33:27

%S 0,1,2,2,4,6,8,4,4,12,16,12,32,24,12,8,64,12,128,24,24,48,256,24,8,96,

%T 8,48,512,20,1024,16,48,192,24,24,2048,384,96,48,4096,40,8192,96,24,

%U 768,16384,48,16,24,192,192,32768,24,48,96,384,1536,65536,40,131072,3072,48,32,96,80,262144,384,768,40,524288,48

%N Xor-Moebius transform of A156552.

%C Unique sequence satisfying SumXOR_{d divides n} a(d) = A156552(n) for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of the Xor-Moebius transform.

%C The ordinary Möbius transform of A156552 is given in A297112.

%C It seems that A091629 gives the fixed points of this sequence.

%H Antti Karttunen, <a href="/A297106/b297106.txt">Table of n, a(n) for n = 1..2048</a>

%o (PARI)

%o A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};

%o A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));

%o A297106(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A156552(d)))); (v); } \\ after code in A295901.

%Y Cf. A064989, A091629, A156552, A295901, A297108, A297112.

%K nonn

%O 1,3

%A _Antti Karttunen_, Dec 25 2017