A369069 - OEIS

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A369069

Dirichlet convolution of Liouville's lambda (A008836) with A083345, where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.

0, 1, 1, 0, 1, 3, 1, 3, 1, 5, 1, 0, 1, 7, 6, -1, 1, 2, 1, 0, 8, 11, 1, 8, 1, 13, 0, 0, 1, 14, 1, 6, 12, 17, 10, 0, 1, 19, 14, 14, 1, 20, 1, 0, 5, 23, 1, -3, 1, 2, 18, 0, 1, 0, 14, 20, 20, 29, 1, 0, 1, 31, 7, -3, 16, 32, 1, 0, 24, 34, 1, 5, 1, 37, 3, 0, 16, 38, 1, -5, 4, 41, 1, 0, 20, 43, 30, 32, 1, 9, 18, 0, 32, 47

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OFFSET

1,6

COMMENTS

In contrast to A347395, this sequence has also nonpositive values after the initial term. It seems that A342090 gives most of the positions of nonpositive terms here, apart from k = 0, 729, 1458, 3645, 5103, 5832, 7290, ..., etc.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

FORMULA

a(n) = Sum_{d|n} A008836(n/d) * A083345(d).

PROG

(PARI)

A008836(n) = ((-1)^bigomega(n));

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

A369069(n) = sumdiv(n, d, A008836(n/d)*A083345(d));

CROSSREFS

Cf. A003415, A008836, A083345, A342090.

Cf. also A347395, A369068.