A143519 Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)*A010051(d); A054525 * A010051.

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, 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, 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

Offset: 1

Author

Gary W. Adamson, Aug 22 2008

Keywords

• sign

Comments

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

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

Examples

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

Table[Sum[MoebiusMu[n/d] Boole[PrimeQ@ d], {d, Divisors@ n}], {n, 89}] (* Michael De Vlieger, Jul 19 2017 *)

PARI

A143519(n) = sumdiv(n,d,isprime(d)*moebius(n/d)); \\ (After Luschny's Sage-code) - Antti Karttunen, Jul 19 2017

Sage

def A143519(n) :
    D = filter(is_prime, divisors(n))
    return add(moebius(n/d) for d in D)
[A143519(n) for n in (1..89)]   # Peter Luschny, Feb 01 2012

Formula

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

Row sums of triangle A143518.

a(n) = Sum_{d|n} A010051(d)*A008683(n/d). - Antti Karttunen, Jul 19 2017

a(n) = Sum_{a*b*c=n} omega(a)*mu(b)*mu(c). - Benedict W. J. Irwin, Mar 02 2022

Extensions

More terms from R. J. Mathar, Jan 19 2009