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A341444 Dirichlet inverse of A083399, where A083399(n) = 1 + omega(n).

%I A341444 #18 Jul 23 2022 09:58:04
%S A341444 1,-2,-2,2,-2,5,-2,-2,2,5,-2,-7,-2,5,5,2,-2,-7,-2,-7,5,5,-2,9,2,5,-2,
%T A341444 -7,-2,-16,-2,-2,5,5,5,14,-2,5,5,9,-2,-16,-2,-7,-7,5,-2,-11,2,-7,5,-7,
%U A341444 -2,9,5,9,5,5,-2,30,-2,5,-7,2,5,-16,-2,-7,5,-16,-2,-23,-2,5,-7,-7,5,-16,-2,-11,2,5,-2,30,5,5,5,9,-2,30,5
%N A341444 Dirichlet inverse of A083399, where A083399(n) = 1 + omega(n).
%C A341444 The Dirichlet inverse function, a(n) = (omega + 1)^(-1)(n). - Original name.
%H A341444 Antti Karttunen, <a href="/A341444/b341444.txt">Table of n, a(n) for n = 1..20000</a>
%H A341444 Carl-Erik Fröberg, <a href="https://doi.org/10.1007/BF01933420">On the prime zeta function</a>, BIT Numerical Mathematics, Vol. 8, No. 3 (1968), pp. 187-202.
%H A341444 H. Hwang and S. Janson, <a href="https://projecteuclid.org/euclid.ejp/1464820181">A central limit theorem for random ordered factorizations of integers</a>, Electron. J. Probab., 16(12):347-361, 2011.
%H A341444 Antti Karttunen, <a href="/A341444/a341444.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H A341444 M. D. Schmidt, <a href="https://arxiv.org/abs/2102.05842">New characterizations of the summatory function of the Moebius function</a>, arXiv:2102.05842 [math.NT], 2021.
%H A341444 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A341444 a(n) = (-1)^A001222(n)*Sum_{d | n} A008683(n/d)^2*A008480(d).
%F A341444 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A083399(n/d) * a(d). - _Antti Karttunen_, Jul 21 2022
%t A341444 a[1] = 1; a[n_] := a[n] = -DivisorSum[n, (PrimeNu[n/#] + 1)*a[#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Jul 21 2022 *)
%o A341444 (PARI) cOmega(n) = if (n==1, 1, my(f=factor(n)); bigomega(n)!*prod(k=1, #f~, 1/f[k,2]!)); \\ A008480
%o A341444 a(n) = (-1)^bigomega(n)*sumdiv(n, d, moebius(n/d)^2*cOmega(d));
%o A341444 (PARI)
%o A341444 memoA341444 = Map();
%o A341444 A341444(n) = if(1==n,1,my(v); if(mapisdefined(memoA341444,n,&v), v, v = -sumdiv(n,d,if(d<n,(1+omega(n/d))*A341444(d),0)); mapput(memoA341444,n,v); (v))); \\ _Antti Karttunen_, Jul 21 2022~
%Y A341444 Dirichlet inverse of A083399.
%Y A341444 Cf. A001221, A001222, A008480, A008683, A008966, A341472 (partial sums).
%Y A341444 Cf. also A334743.
%K A341444 sign
%O A341444 1,2
%A A341444 _Michel Marcus_, Feb 12 2021
%E A341444 Data section extended up to a(91) and name edited by _Antti Karttunen_, Jul 21 2022