A137851 a(n) = A054525(n) * A061397(n).

0, 2, 3, -2, 5, -5, 7, 0, -3, -7, 11, 2, 13, -9, -8, 0, 17, 3, 19, 2, -10, -13, 23, 0, -5, -15, 0, 2, 29, 10, 31, 0, -14, -19, -12, 0, 37, -21, -16, 0, 41, 12, 43, 2, 3, -25, 47, 0, -7, 5, -20, 2, 53, 0, -16, 0, -22, -31, 59, -2, 61, -33, 3, 0, -18, 16, 67, 2, -26, 14, 71, 0, 73, -39, 5, 2, -18, 18, 79, 0, 0, -43, 83, -2, -22, -45, -32, 0

Offset: 1

Author

Gary W. Adamson, Feb 14 2008

Keywords

• easy
• sign

Comments

Equals row sums of triangle A143517. - Gary W. Adamson, Aug 22 2008

Examples

			a(4) = -2 = (0, -1, 0, 1) dot (0, 2, 3, 0), where (0, -1, 0, 1) = row 4 of the Möbius triangle A054525 and (0, 2, 3, 0) = the first 4 terms of A061397.
		

Crossrefs

Cf. A061397, A054525, A143517, A143519.

Programs

Maple

A061397 := proc(n) if isprime(n) then n; else 0 ; fi ; end: A054525 := proc(n,k) if n mod k = 0 then numtheory[mobius](n/k); else 0; fi ; end: A137851 := proc(n) local k ; add(A061397(k)* A054525(n,k),k=1..n) ; end: seq(A137851(n),n=1..120) ; # R. J. Mathar, May 23 2008

Mathematica

a[n_] := If[n == 1, 0, With[{p = FactorInteger[n][[All, 1]]}, p*MoebiusMu[n/p] // Total]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 13 2023 *)

Sage

def A137851(n):
    return add(d*moebius(n//d) for d in divisors(n) if is_prime(d))
[A137851(n) for n in (1..88)] # Peter Luschny, Feb 01 2012

Formula

A054525 * A061397 = Möbius transform of [0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, ...].

Dirichlet g.f.: primezeta(s-1)/zeta(s). - Benedict W. J. Irwin, Jul 11 2018

a(n) = Sum_{p|n} p*mu(n/p), where p is prime. - Ridouane Oudra, Nov 12 2019

Extensions

More terms from R. J. Mathar, May 23 2008