A062778 Values of Moebius-transform of PrimePi function.

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, 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, 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

Offset: 1

Author

Labos Elemer, Jul 18 2001

Keywords

• nonn

Examples

			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.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A007444, A007445.

Programs

Mathematica

f[n_] := Block[{d = Divisors@n}, Plus @@ (MoebiusMu /@ (n/d)*PrimePi /@ d)]; Array[f, 94] (* Robert G. Wilson v, Dec 07 2005 *)

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

PARI

a(n) = sumdiv(n, d, primepi(d)*moebius(n/d)); \\ Michel Marcus, Nov 05 2018

Formula

a(n) = Sum_{d|n} pi(n/d)*mu(d).