A319443 Number of distinct Eisenstein primes in the factorization of n.

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 3, 2, 1, 1, 2, 2, 2, 3, 2, 1, 2, 1, 3, 1, 3, 1, 3, 2, 1, 2, 2, 3, 2, 2, 3, 3, 2, 1, 4, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 3, 2, 1, 3, 2, 3, 3, 1, 3, 3, 2, 2, 2, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 2, 1, 2, 1, 4, 2, 3, 2

Offset: 1

Author

Jianing Song, Sep 19 2018

Keywords

• nonn

Comments

Equivalent of omega (A001221) in the ring of Eisenstein integers.

z is an Eisenstein prime iff z has prime norm or z is the product of a rational prime congruent to 2 modulo 3 and an Eisenstein unit (one of +-1 or (+-1 +- sqrt(3)*i)/2).

Associated Eisenstein prime divisors are counted only once.

Let s(n) be the smallest k with a(k) = n, then we have: s(0) = 1, s(1) = 2, s(2) = 6, s(2n-1) = 2*A121940(n-1), s(2n) = 6*A121940(n-1).

Examples

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
Over the Gaussian integers, 5187 = 3*7*13*19 is factored as w'*(1 + w)^2*(2 + w)*(2 + w')*(3 + w)*(3 + w')*(3 + 2w)*(3 + 2w'), the distinct Eisenstein prime factors are 1 + w, 2 + w, 2 + w', 3 + w, 3 + w', 3 + 2w and 3 + 2w', so a(5187) = 7.
Over the Gaussian integers, 1006655265000 = 2^3*3^2*5^4*7^5*11^3 is factored as w'^2*(1 + w)^4*2^3*(2 + w)*(2 + w')*5^4*11^3, the distinct Eisenstein prime factors are 1 + w, 2, 2 + w, 2 + w', 5 and 11, so a(1006655265000) = 6.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Wikipedia, Eisenstein integer

Crossrefs

Cf. A121940.

Equivalent of arithmetic functions in the ring of Eisenstein integers (the corresponding functions in the ring of integers are in the parentheses): A319442 ("d", A000005), A319449 ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), this sequence ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).

Equivalent in the ring of Gaussian integers: A086275.

Programs

Mathematica

f[p_, e_] := If[Mod[p, 3] == 1, 2, 1]; eisOmega[1] = 0; eisOmega[n_] := Plus @@ f @@@ FactorInteger[n]; Array[eisOmega, 100] (* Amiram Eldar, Feb 10 2020 *)

PARI

a(n)=my(f=factor(n)[, 1]); sum(i=1, #f, if(f[i]%3==1, 2, 1))

Formula

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