OFFSET
1,2
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.
FORMULA
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A083399(n/d) * a(d). - Antti Karttunen, Jul 21 2022
MATHEMATICA
a[1] = 1; a[n_] := a[n] = -DivisorSum[n, (PrimeNu[n/#] + 1)*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)
PROG
(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(d<n, (1+omega(n/d))*A341444(d), 0)); mapput(memoA341444, n, v); (v))); \\ Antti Karttunen, Jul 21 2022~
CROSSREFS
KEYWORD
sign
AUTHOR
Michel Marcus, Feb 12 2021
EXTENSIONS
Data section extended up to a(91) and name edited by Antti Karttunen, Jul 21 2022
STATUS
approved