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Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)*A010051(d); A054525 * A010051.
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%I #41 Mar 18 2022 00:30:22

%S 0,1,1,-1,1,-2,1,0,-1,-2,1,1,1,-2,-2,0,1,1,1,1,-2,-2,1,0,-1,-2,0,1,1,

%T 3,1,0,-2,-2,-2,0,1,-2,-2,0,1,3,1,1,1,-2,1,0,-1,1,-2,1,1,0,-2,0,-2,-2,

%U 1,-1,1,-2,1,0,-2,3,1,1,-2,3,1,0,1,-2,1,1,-2,3,1,0,0,-2,1,-1,-2,-2,-2,0,1

%N Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)*A010051(d); A054525 * A010051.

%C A010051 = A051731 * A143519 (since A051731 = the inverse Mobius transform).

%C A000720(n) = Sum_{k=1..n} a(k) floor(n/k) where A000720(n) is the number of primes <= n. - _Steven Foster Clark_, May 25 2018

%H Antti Karttunen, <a href="/A143519/b143519.txt">Table of n, a(n) for n = 1..10000</a>

%F Mobius transform of A010051, the characteristic function of the primes.

%F Row sums of triangle A143518.

%F a(n) = Sum_{d|n} A010051(d)*A008683(n/d). - _Antti Karttunen_, Jul 19 2017

%F a(n) = Sum_{a*b*c=n} omega(a)*mu(b)*mu(c). - _Benedict W. J. Irwin_, Mar 02 2022

%e a(4) = -1 since row 4 of triangle A043518 = (0, -1, 0, 0).

%e a(4) = -1 = (0, -1, 0, 1) dot (0, 1, 1, 0), where (0, -1, 0, 1) = row 4 of A054525 and A010051 = (0, 1, 1, 0, 1, 0, 1, 0, ...).

%t Table[Sum[MoebiusMu[n/d] Boole[PrimeQ@ d], {d, Divisors@ n}], {n, 89}] (* _Michael De Vlieger_, Jul 19 2017 *)

%o (Sage)

%o def A143519(n) :

%o D = filter(is_prime, divisors(n))

%o return add(moebius(n/d) for d in D)

%o [A143519(n) for n in (1..89)] # _Peter Luschny_, Feb 01 2012

%o (PARI) A143519(n) = sumdiv(n,d,isprime(d)*moebius(n/d)); \\ (After Luschny's Sage-code) - _Antti Karttunen_, Jul 19 2017

%Y Cf. A008683, A010051, A143518, A054525, A137851.

%K sign

%O 1,6

%A _Gary W. Adamson_, Aug 22 2008

%E More terms from _R. J. Mathar_, Jan 19 2009