A319444 - OEIS

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

%S 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,

%T 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,

%U 3,4,1,7,2,3,4,4,3,5,2,5,8,2,1,6,2,3,3

%N Total number of factors in a factorization of n into Eisenstein primes.

%C Equivalent of Omega (A001222) 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 The smallest k with a(k) = n is A038754(n).

%H Jianing Song, <a href="/A319444/b319444.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 Completely additive with a(p) = 2 if p = 3 or p == 1 (mod 3) and a(p) = 1 if p == 2 (mod 3).

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

%e 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.

%e 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)'.

%e 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.

%t 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 *)

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

%Y Cf. A038754.

%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), A319443 ("omega", A001221), this sequence ("Omega", A001222), A319448 ("mu", A008683).

%Y Equivalent in the ring of Gaussian integers: A078458.

%K nonn

%O 1,3

%A _Jianing Song_, Sep 19 2018