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A319444
Total number of factors in a factorization of n into Eisenstein primes.
7
0, 1, 2, 2, 1, 3, 2, 3, 4, 2, 1, 4, 2, 3, 3, 4, 1, 5, 2, 3, 4, 2, 1, 5, 2, 3, 6, 4, 1, 4, 2, 5, 3, 2, 3, 6, 2, 3, 4, 4, 1, 5, 2, 3, 5, 2, 1, 6, 4, 3, 3, 4, 1, 7, 2, 5, 4, 2, 1, 5, 2, 3, 6, 6, 3, 4, 2, 3, 3, 4, 1, 7, 2, 3, 4, 4, 3, 5, 2, 5, 8, 2, 1, 6, 2, 3, 3
OFFSET
1,3
COMMENTS
Equivalent of Omega (A001222) 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).
The smallest k with a(k) = n is A038754(n).
LINKS
FORMULA
Completely additive with a(p) = 2 if p = 3 or p == 1 (mod 3) and a(p) = 1 if p == 2 (mod 3).
EXAMPLE
Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
a(54) = a(2*3^3) = 1*a(2) + 3*a(3) = 1*1 + 3*2 = 7. Over the Gaussian integers, 54 is factored as -2*(1 + w)^6.
a(63) = a(3^2*7) = 2*a(3) + 1*a(7) = 2*2 + 1*2 = 6. Over the Gaussian integers, 63 is factored as w'^2*(1 + w)^4*(2 + w)*(2 + w)'.
a(1006655265000) = a(2^3*3^2*5^4*7^5*11^3) = 3*a(2) + 2*a(3) + 4*a(5) + 5*a(7) + 3*a(11) = 3*1 + 2*2 + 4*1 + 5*2 + 3*1 = 24. Over the Gaussian integers, 1006655265000 is factored as w'^2*(1 + w)^4*2^3*(2 + w)*(2 + w')*5^4*11^3.
MATHEMATICA
f[p_, e_] := e * If[Mod[p, 3] == 2, 1, 2]; eisBigomega[1] = 0; eisBigomega[n_] := Plus @@ f @@@ FactorInteger[n]; Array[eisBigomega, 100] (* Amiram Eldar, Feb 10 2020 *)
PROG
(PARI) a(n)=my(f=factor(n)); sum(i=1, #f~, if(f[i, 1]%3==2, 1, 2)*f[i, 2])
CROSSREFS
Cf. A038754.
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), A319443 ("omega", A001221), this sequence ("Omega", A001222), A319448 ("mu", A008683).
Equivalent in the ring of Gaussian integers: A078458.
Sequence in context: A333867 A030717 A280716 * A071285 A289438 A008678
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 19 2018
STATUS
approved