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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 (list; graph; refs; listen; history; text; internal format)
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(2n-1) = 2*A121940(n-1), s(2n) = 6*A121940(n-1).

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

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Last modified January 18 16:27 EST 2020. Contains 331011 sequences. (Running on oeis4.)