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A349434
Dirichlet convolution of A129283 (n + its arithmetic derivative) with A349337 (Dirichlet inverse of A230593).
5
1, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, -2, 0, 0, 0, 6, 0, -3, 0, -2, 0, 0, 0, 0, 5, 0, 6, -2, 0, 0, 0, 10, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, -2, -3, 0, 0, -6, 7, -5, 0, -2, 0, -3, 0, 0, 0, 0, 0, 4, 0, 0, -3, 22, 0, 0, 0, -2, 0, 0, 0, -5, 0, 0, -5, -2, 0, 0, 0, -6, 21, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, -2, 0, 0, 0, -4, 0, -7, -3, 7
OFFSET
1,4
COMMENTS
Dirichlet convolution of this sequence with A349338 is A348976.
LINKS
FORMULA
a(n) = Sum_{d|n} A129283(n/d) * A349337(d).
MATHEMATICA
s[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); a[n_] := DivisorSum[n, sinv[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
PROG
(PARI)
up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129283(n) = (n+A003415(n));
A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v349337 = DirInverseCorrect(vector(up_to, n, A230593(n)));
A349337(n) = v349337[n];
A349434(n) = sumdiv(n, d, A129283(n/d)*A349337(d));
CROSSREFS
Cf. A003415, A129283, A230593, A349337, A349435 (Dirichlet inverse), A349436 (sum with it).
Cf. also A348976, A349338.
Sequence in context: A057108 A349435 A063958 * A126164 A340317 A145007
KEYWORD
sign
AUTHOR
Antti Karttunen, Nov 17 2021
STATUS
approved