A368647 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.

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, 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, 1, 0, 2, -1, 2, 1, 1, 1, 1, -1, 0, 1, 0, 0, 0, -1, 2, 0, 2

Offset: 1

Author

Amiram Eldar, Jan 02 2024

Keywords

• sign
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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]

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)));}

Formula

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

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

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