

A319443


Number of distinct Eisenstein primes in the factorization of n.


7



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
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OFFSET

1,6


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(2n1) = 2*A121940(n1), s(2n) = 6*A121940(n1).


LINKS

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


FORMULA

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


EXAMPLE

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.


PROG

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


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.
Sequence in context: A236479 A116514 A124767 * A130633 A266499 A226621
Adjacent sequences: A319440 A319441 A319442 * A319444 A319445 A319446


KEYWORD

nonn


AUTHOR

Jianing Song, Sep 19 2018


STATUS

approved



