A368647 - OEIS

%I #7 Jan 02 2024 02:44:08

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

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

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

%N The number of distinct primes of the form 3*k+2 dividing n minus the number of distinct primes of the form 3*k+1 dividing n.

%H Amiram Eldar, <a href="/A368647/b368647.txt">Table of n, a(n) for n = 1..10000</a>

%F Additive with a(p^e) = 0 if p = 3, 1 if p == 2 (mod 3), and -1 if p == 1 (mod 3).

%F a(n) = A005090(n) - A005088(n).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A086241 = 0.641944... .

%t f[p_, e_] := Switch[Mod[p, 3], 0, 0, 1, -1, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(p = factor(n)[, 1]); sum(i = 1, #p, if(p[i]%3 == 0, 0, if(p[i]%3 == 1, -1, 1)));}

%Y Cf. A002476, A003627, A005094, A005088, A005090, A086241.

%K sign,easy

%O 1,10

%A _Amiram Eldar_, Jan 02 2024