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A297108 If n is prime(k)^e, e >= 1, then a(n) = 2^(k-1), otherwise 0; Möbius transform of A048675.

%I A297108 #15 Dec 26 2017 17:33:16
%S A297108 0,1,2,1,4,0,8,1,2,0,16,0,32,0,0,1,64,0,128,0,0,0,256,0,4,0,2,0,512,0,
%T A297108 1024,1,0,0,0,0,2048,0,0,0,4096,0,8192,0,0,0,16384,0,8,0,0,0,32768,0,
%U A297108 0,0,0,0,65536,0,131072,0,0,1,0,0,262144,0,0,0,524288,0,1048576,0,0,0,0,0,2097152,0,2,0,4194304
%N A297108 If n is prime(k)^e, e >= 1, then a(n) = 2^(k-1), otherwise 0; Möbius transform of A048675.
%C A297108 This is also Xor-Moebius transform of A248663, in other words, the unique sequence satisfying SumXOR_{d divides n} a(d) = A248663(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 this transform.
%H A297108 Antti Karttunen, <a href="/A297108/b297108.txt">Table of n, a(n) for n = 1..1024</a>
%F A297108 If A001221(n) = 1 [when n is in A000961], then a(n) = 2^(A297109(n)-1) =  2^(A055396(n)-1), otherwise 0.
%F A297108 a(n) = Sum_{d|n} A048675(d)*A008683(n/d).
%o A297108 (PARI)
%o A297108 A297108(n) = if(1==omega(n),2^(primepi(factor(n)[1,1])-1),0);
%o A297108 \\ A more complicated way which demonstrates the Moebius transform:
%o A297108 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ This function after _Michel Marcus_
%o A297108 A297108(n) = sumdiv(n,d,moebius(n/d)*A048675(d));
%o A297108 \\ And yet another way demonstrating the comment:
%o A297108 A248663(n) = A048675(core(n));
%o A297108 A297108(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A248663(d)))); (v); } \\ after code in A295901.
%Y A297108 Cf. A008683, A048675, A248663, A295901, A297106, A297109.
%K A297108 nonn,base
%O A297108 1,3
%A A297108 _Antti Karttunen_, Dec 25 2017