A319443 - OEIS

%I #19 Feb 10 2020 17:40:35

%S 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,

%T 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,

%U 2,4,1,2,2,3,2,3,3,4,2,2,1,2,1,4,2,3,2

%N Number of distinct Eisenstein primes in the factorization of n.

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

%C 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).

%C Associated Eisenstein prime divisors are counted only once.

%C 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).

%H Jianing Song, <a href="/A319443/b319443.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>

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

%e Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.

%e 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.

%e 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.

%t 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 *)

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

%Y Cf. A121940.

%Y 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).

%Y Equivalent in the ring of Gaussian integers: A086275.

%K nonn

%O 1,6

%A _Jianing Song_, Sep 19 2018