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A359773
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Dirichlet inverse of A356163, where A356163 is the characteristic function of the numbers with an even sum of prime factors (counted with multiplicity).
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11
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1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,225
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COMMENTS
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a(225) = 2 is the first term with absolute value larger than 1.
As A356163 is not multiplicative, neither is this sequence.
For all numbers n with an odd number of odd prime factors (with mult.), a(n) = 0. Proof: Numbers with an odd number of odd prime factors is sequence A335657 (equal to numbers whose odd part is in A067019). In the convolution formula, when n is any term of A335657, either the divisor (n/d) or d (but not both) is also a term of A335657. As A356163 is zero for all A335657, it is easy to show by induction that also a(n) is zero for all such numbers.
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LINKS
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FORMULA
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a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A356163(n/d) * a(d).
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PROG
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(PARI)
A356163(n) = (1-(((n=factor(n))[, 1]~*n[, 2])%2)); \\ After code in A001414.
memoA359773 = Map();
A359773(n) = if(1==n, 1, my(v); if(mapisdefined(memoA359773, n, &v), v, v = -sumdiv(n, d, if(d<n, A356163(n/d)*A359773(d), 0)); mapput(memoA359773, n, v); (v)));
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CROSSREFS
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Cf. A001414, A036347, A036348, A036349, A067019, A335657, A356163, A359774 (parity of terms), A359775 (positions of odd terms), A359776 (of even terms), A359777.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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