|
%I #24 Sep 18 2023 02:02:17
%S 1,0,-3,0,-5,0,-7,0,0,0,-11,0,-13,0,15,0,-17,0,-19,0,21,0,-23,0,0,0,0,
%T 0,-29,0,-31,0,33,0,35,0,-37,0,39,0,-41,0,-43,0,0,0,-47,0,0,0,51,0,
%U -53,0,55,0,57,0,-59,0,-61,0,0,0,65,0,-67,0,69,0,-71,0,-73,0,0,0,77,0,-79,0,0,0,-83,0,85,0,87,0,-89
%N Dirichlet inverse of A193356, which is defined as n if n is odd, 0 if n is even.
%H Antti Karttunen, <a href="/A349343/b349343.txt">Table of n, a(n) for n = 1..20000</a>
%F a(2n) = 0, a(2n+1) = A349341(2n+1) for all n >= 1.
%F Multiplicative with a(p^e) = 0 if p=2 or e>1, otherwise a(p) = -p. - (After _Sebastian Karlsson_'s similar formula for A349341).
%t a[1]=1;a[n_]:=-DivisorSum[n,If[OddQ[n/#],n/#,0]*a@#&,#<n&];Array[a,89] (* _Giorgos Kalogeropoulos_, Nov 15 2021 *)
%t f[p_, e_] := If[e == 1, -p, 0]; f[2, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 18 2023 *)
%o (PARI) A349343(n) = { my(f = factor(n)); prod(i=1, #f~, if((2==f[i,1])||(f[i,2]>1), 0, -f[i,1])); };
%Y Agrees with A349341 on odd numbers.
%Y Cf. A193356, and also A328203, A349134, A349344, A349346, A349353.
%K sign,easy,mult
%O 1,3
%A _Antti Karttunen_, Nov 15 2021
|
