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A297108
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If n is prime(k)^e, e >= 1, then a(n) = 2^(k-1), otherwise 0; Möbius transform of A048675.
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9
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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, 1024, 1, 0, 0, 0, 0, 2048, 0, 0, 0, 4096, 0, 8192, 0, 0, 0, 16384, 0, 8, 0, 0, 0, 32768, 0, 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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..1024
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FORMULA
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If A001221(n) = 1 [when n is in A000961], then a(n) = 2^(A297109(n)-1) = 2^(A055396(n)-1), otherwise 0.
a(n) = Sum_{d|n} A048675(d)*A008683(n/d).
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PROG
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(PARI)
A297108(n) = if(1==omega(n), 2^(primepi(factor(n)[1, 1])-1), 0);
\\ A more complicated way which demonstrates the Moebius transform:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ This function after Michel Marcus
A297108(n) = sumdiv(n, d, moebius(n/d)*A048675(d));
\\ And yet another way demonstrating the comment:
A248663(n) = A048675(core(n));
A297108(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A248663(d)))); (v); } \\ after code in A295901.
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CROSSREFS
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Cf. A008683, A048675, A248663, A295901, A297106, A297109.
Sequence in context: A348508 A077954 A077979 * A307626 A122161 A067164
Adjacent sequences: A297105 A297106 A297107 * A297109 A297110 A297111
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KEYWORD
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nonn,base
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AUTHOR
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Antti Karttunen, Dec 25 2017
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STATUS
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approved
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