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

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, -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, -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

Offset: 1

Author

Michel Marcus, Feb 12 2021

Keywords

• sign

Comments

The Dirichlet inverse function, a(n) = (omega + 1)^(-1)(n). - Original name.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Carl-Erik Fröberg, On the prime zeta function, BIT Numerical Mathematics, Vol. 8, No. 3 (1968), pp. 187-202.
• H. Hwang and S. Janson, A central limit theorem for random ordered factorizations of integers, Electron. J. Probab., 16(12):347-361, 2011.
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
• M. D. Schmidt, New characterizations of the summatory function of the Moebius function, arXiv:2102.05842 [math.NT], 2021.
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Dirichlet inverse of A083399.

Cf. A001221, A001222, A008480, A008683, A008966, A341472 (partial sums).

Cf. also A334743.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = -DivisorSum[n, (PrimeNu[n/#] + 1)*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)

PARI

cOmega(n) = if (n==1, 1, my(f=factor(n)); bigomega(n)!*prod(k=1, #f~, 1/f[k,2]!)); \\ A008480
a(n) = (-1)^bigomega(n)*sumdiv(n, d, moebius(n/d)^2*cOmega(d));

PARI

memoA341444 = Map();
A341444(n) = if(1==n,1,my(v); if(mapisdefined(memoA341444,n,&v), v, v = -sumdiv(n,d,if(dA341444(d),0)); mapput(memoA341444,n,v); (v))); \\ Antti Karttunen, Jul 21 2022~

Formula

a(n) = (-1)^A001222(n)*Sum_{d | n} A008683(n/d)^2*A008480(d).

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083399(n/d) * a(d). - Antti Karttunen, Jul 21 2022

Extensions

Data section extended up to a(91) and name edited by Antti Karttunen, Jul 21 2022