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%I #16 Nov 05 2018 06:32:30
%S 0,1,2,1,3,0,4,2,2,0,5,1,6,1,1,2,7,2,8,3,2,2,9,2,6,2,5,2,10,3,11,5,4,
%T 3,4,2,12,3,4,2,13,3,14,5,6,4,15,4,11,5,6,5,16,4,8,5,6,5,17,2,18,6,8,
%U 7,9,4,19,7,8,6,20,5,21,8,9,8,12,6,22,8,13,8,23,6,13,8,11,7,24,4,14,9,11,8
%N Values of Moebius-transform of PrimePi function.
%H Harry J. Smith, <a href="/A062778/b062778.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = Sum_{d|n} pi(n/d)*mu(d).
%e n=12, divisors = D(12) = {1,2,3,4,6,12}, pi(12/divisors) = {5,3,2,2,1,0}, mu(divisors) = {1,-1,-1,0,1,0}, Sum = 5*1 - 3*1 - 2*1 + 0 + 1*1 + 0 = 1, thus a(12)=1; for p=prime(n), pi(p/divisor) = {n,0}, mu({1,p})={1,-1}, Sum = 1*n + 0 = n, so a(prime(n)) = n.
%t f[n_] := Block[{d = Divisors@n}, Plus @@ (MoebiusMu /@ (n/d)*PrimePi /@ d)]; Array[f, 94] (* _Robert G. Wilson v_, Dec 07 2005 *)
%o (PARI) { for (n=1, 1000, d=divisors(n); write("b062778.txt", n, " ", sum(k=1, length(d), primepi(n/d[k]) * moebius(d[k]))) ) } \\ _Harry J. Smith_, Aug 10 2009
%o (PARI) a(n) = sumdiv(n, d, primepi(d)*moebius(n/d)); \\ _Michel Marcus_, Nov 05 2018
%Y Cf. A007444, A007445.
%K nonn
%O 1,3
%A _Labos Elemer_, Jul 18 2001
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