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%I #27 Jun 13 2023 05:13:18
%S 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,
%T -15,0,2,29,10,31,0,-14,-19,-12,0,37,-21,-16,0,41,12,43,2,3,-25,47,0,
%U -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
%N a(n) = A054525(n) * A061397(n).
%C Equals row sums of triangle A143517. - _Gary W. Adamson_, Aug 22 2008
%F A054525 * A061397 = Möbius transform of [0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, ...].
%F Dirichlet g.f.: primezeta(s-1)/zeta(s). - _Benedict W. J. Irwin_, Jul 11 2018
%F a(n) = Sum_{p|n} p*mu(n/p), where p is prime. - _Ridouane Oudra_, Nov 12 2019
%e 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.
%p 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
%t a[n_] := If[n == 1, 0, With[{p = FactorInteger[n][[All, 1]]}, p*MoebiusMu[n/p] // Total]];
%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jun 13 2023 *)
%o (Sage)
%o def A137851(n):
%o return add(d*moebius(n//d) for d in divisors(n) if is_prime(d))
%o [A137851(n) for n in (1..88)] # _Peter Luschny_, Feb 01 2012
%Y Cf. A061397, A054525, A143517, A143519.
%K easy,sign
%O 1,2
%A _Gary W. Adamson_, Feb 14 2008
%E More terms from _R. J. Mathar_, May 23 2008
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