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A359763
Dirichlet inverse of A065043, where A065043 is the characteristic function of the numbers with an even number of prime factors (counted with multiplicity).
13
1, 0, 0, -1, 0, -1, 0, 0, -1, -1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, -1, 0, 1, -1, -1, 0, 0, 0, 0, 0, 0, -1, -1, -1, 2, 0, -1, -1, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0, 0, 1, -1, 1, -1, -1, 0, 3, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 3, -1, -1, -1, 1, 0, 3, -1, 0, -1, -1, -1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0
OFFSET
1,36
COMMENTS
As A065043 is not multiplicative, neither is this sequence.
For all numbers n with an odd number of prime factors (with mult.), a(n) = 0. Proof: In the convolution formula, when n is any term of A026424, either the divisor (n/d) or d (but not both) has an odd number of prime factors. As A065043 is zero for all A026424, it is easy to show by induction that also a(n) is zero for all such numbers. Therefore, nonzero values (including any odd values, see A359765) occur only on a subset of A028260, and A359764(n) <= A065043(n).
FORMULA
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A065043(n/d) * a(d).
a(n) = A359773(A003961(n)) = A359780(A003961(n)) = A359823(A003961(n)).
a(n) = a(A046523(n)) for all n, i.e., the result depends only on the prime signature of n, A101296.
PROG
(PARI)
A065043(n) = (1 - (bigomega(n)%2));
memoA359763 = Map();
A359763(n) = if(1==n, 1, my(v); if(mapisdefined(memoA359763, n, &v), v, v = -sumdiv(n, d, if(d<n, A065043(n/d)*A359763(d), 0)); mapput(memoA359763, n, v); (v)));
CROSSREFS
Cf. A003961, A026424, A028260, A046523, A065043, A101296, A359764 (parity of the terms), A359765 (positions of odd terms), A359766 (of even terms), A359767.
Cf. also A358777, A359773, A359780, A359823 for similar constructions and A008966 which is Dirichlet inverse of A065043(n)-A066829(n) = A008836(n).
Sequence in context: A309163 A320655 A359786 * A277017 A178498 A353422
KEYWORD
sign
AUTHOR
Antti Karttunen, Jan 13 2023
STATUS
approved