A319449 -id:A319449 - cp's OEIS frontend

A319442 Number of divisors of n over the Eisenstein integers.

1, 2, 3, 3, 2, 6, 4, 4, 5, 4, 2, 9, 4, 8, 6, 5, 2, 10, 4, 6, 12, 4, 2, 12, 3, 8, 7, 12, 2, 12, 4, 6, 6, 4, 8, 15, 4, 8, 12, 8, 2, 24, 4, 6, 10, 4, 2, 15, 9, 6, 6, 12, 2, 14, 4, 16, 12, 4, 2, 18, 4, 8, 20, 7, 8, 12, 4, 6, 6, 16, 2, 20, 4, 8, 9, 12, 8, 24, 4, 10

Offset: 1

Jianing Song, Sep 19 2018

Equivalent of d (A000005) in the ring of Eisenstein integers.

Divisors which are associates are identified (two Eisenstein integers z1, z2 are associates if z1 = u * z2 where u is an Eisenstein unit, i.e., one of +-1 or (+-1 +- sqrt(3)*i)/2).

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
Divisors of 7 over the Eisenstein integers are 1, 2 + w, 2 + w', 7 and their association, so a(7) = 4.
Divisors of 9 over the Eisenstein integers are 1, 1 + w, 3, 3 + 3w, 9 and their association, so a(9) = 5.

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

Equivalent of arithmetic functions in the ring of Eisenstein integers (the corresponding functions in the ring of integers are in the parentheses): this sequence ("d", A000005), A319449 ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).

Equivalent in the ring of Gaussian integers: A062327.

A319442 := proc(n) local t, f, j, e, m; t := 1: f := ifactors(n)[2];
   for j from 1 to nops(f) do
      e := f[j, 2] + 1; m := f[j, 1] mod 3;
      if   m = 0 then 2*e-1
      elif m = 1 then e^2
      else e fi;
      t := t * % od;
t end: seq(A319442(n), n=1..80); # Peter Luschny, Oct 03 2018

f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisNumDiv, 100] (* Amiram Eldar, Feb 10 2020 *)

A319442(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==3, r*=(2*e+1));
        if(p%3==1, r*=(e+1)^2);
        if(p%3==2, r*=(e+1));
    );
    return(r);
}

Multiplicative with a(3^e) = 2*e + 1, a(p^e) = (e + 1)^2 if p == 1 (mod 3) and e + 1 if p == 2 (mod 3).

A319445 Number of Eisenstein integers in a reduced system modulo n.

Original entry on oeis.org

1, 3, 6, 12, 24, 18, 36, 48, 54, 72, 120, 72, 144, 108, 144, 192, 288, 162, 324, 288, 216, 360, 528, 288, 600, 432, 486, 432, 840, 432, 900, 768, 720, 864, 864, 648, 1296, 972, 864, 1152, 1680, 648, 1764, 1440, 1296, 1584, 2208, 1152, 1764, 1800, 1728, 1728, 2808

Offset: 1

Jianing Song, Sep 19 2018

Equivalent of phi (A000010) in the ring of Eisenstein integers.
Number of units in the ring Z[w]/nZ[w], where Z[w] is the ring of Eisenstein integers.
a(n) is the number of elements in G(n) = {a + b*w: a, b in Z/nZ and gcd(a^2 + a*b + b^2, n) = 1} where w = (1 + sqrt(3)*i)/2.
a(n) is the number of ordered pairs (a, b) modulo n such that gcd(a^2 + a*b + b^2, n) = 1.
For n > 2, a(n) is divisible by 6.

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
{1, w, w'} is the set of 3 units in the Eisenstein integers modulo 2, so a(2) = 3.
{1, w, w^2, -1, w', w'^2} is the set of 6 units in the Eisenstein integers modulo 3, so a(3) = 6.
{1, w, w'} is the set of 3 units in the Eisenstein integers modulo 2, so a(2) = 3.
{1, w, 1 + w, w', 1 + w', -1 + 2w, -1, -w, -1 - w, -w', -1 - w', -1 + 2w'} is the set of 12 units in the Eisenstein integers modulo 4, so a(4) = 12.

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

Cf. A007434.
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), this sequence ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).
Equivalent in the ring of Gaussian integers: A079458.

f[p_, e_] := If[p == 3 , 2*3^(2*e - 1), Switch[Mod[p, 3], 1, (p - 1)^2*p^(2*e - 2), 2, (p^2 - 1)*p^(2*e - 2)]]; eisPhi[1] = 1; eisPhi[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisPhi, 100] (* Amiram Eldar, Feb 10 2020 *)

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==3, r*=2*3^(2*e-1));
        if(p%3==1, r*=(p-1)^2*p^(2*e-2));
        if(p%3==2, r*=(p^2-1)*p^(2*e-2));
    );
    return(r);
}

Multiplicative with a(3^e) = 2*3^(2*e-1), a(p^e) = phi(p^e)^2 = (p-1)^2*p^(2*e-2) if p == 1 (mod 3) and J_2(p^e) = A007434(p^e) = (p^2 - 1)*p^(2*e-2) if p == 2 (mod 3).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (8/27) * Product_{p prime == 1 (mod 3)} (1 - 2/p^2 + 1/p^3) * Product_{p prime == 2 (mod 3)} (1 - 1/p^3) = 0.2410535987... . - Amiram Eldar, Feb 13 2024

A317797 Sum of the norm of divisors of n over Gaussian integers, with associated divisors counted only once.

Original entry on oeis.org

1, 7, 10, 31, 36, 70, 50, 127, 91, 252, 122, 310, 196, 350, 360, 511, 324, 637, 362, 1116, 500, 854, 530, 1270, 961, 1372, 820, 1550, 900, 2520, 962, 2047, 1220, 2268, 1800, 2821, 1444, 2534, 1960, 4572, 1764, 3500, 1850, 3782, 3276, 3710, 2210, 5110, 2451, 6727

Offset: 1

Jianing Song, Aug 07 2018

Equivalent of sigma (A000203) in the ring of Gaussian integers. Note that only norms are summed up.

			Let ||d|| denote the norm of d.
a(2) = ||1|| + ||1 + i|| + ||2|| = 1 + 2 + 4 = 7.
a(5) = ||1|| + ||2 + i|| + ||2 - i|| + ||5|| = 1 + 5 + 5 + 25 = 36. Note that 2 - i and 1 + 2i are associated so their norm is only counted once.

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

Cf. A001157.
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), this sequence ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319449.

f[p_, e_] := If[p == 2, 2^(2*e + 1) - 1, Switch[Mod[p, 4], 1, ((p^(e + 1) - 1)/(p - 1))^2, 3, (p^(2 e + 2) - 1)/(p^2 - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 12 2020 *)

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2, r*=(2^(2*e+1)-1));
        if(Mod(p,4)==1, r*=((p^(e+1)-1)/(p-1))^2);
        if(Mod(p,4)==3, r*=(p^(2*e+2)-1)/(p^2-1));
    );
    return(r);
}

Multiplicative with a(2^e) = sigma(2^(2e)) = 2^(2e+1) - 1, a(p^e) = sigma(p^e)^2 = ((p^(e+1) - 1)/(p - 1))^2 if p == 1 (mod 4) and sigma_2(p^e) = A001157(p^e) = (p^(2e+2) - 1)/(p^2 - 1) if p == 3 (mod 4).

A319446 Exponent of the group of the Eisenstein integers in a reduced system modulo n.

Original entry on oeis.org

1, 3, 6, 6, 24, 6, 6, 12, 6, 24, 120, 6, 12, 6, 24, 24, 288, 6, 18, 24, 6, 120, 528, 12, 120, 12, 18, 6, 840, 24, 30, 48, 120, 288, 24, 6, 36, 18, 12, 24, 1680, 6, 42, 120, 24, 528, 2208, 24, 42, 120, 288, 12, 2808, 18, 120, 12, 18, 840, 3480, 24, 60

Offset: 1

Jianing Song, Sep 19 2018

nonn

Equivalent of psi (A002322) in the ring of Eisenstein integers.
a(n) is the exponent of the multiplicative group of Eisenstein integers modulo n, i.e., (Z[w]/nZ[w])* = {a + b*w : a, b in Z/nZ and gcd(a^2 + a*b + b^2, n) = 1} where w = (1 + sqrt(3)*i)/2. The number of elements in (Z[w]/nZ[w])* is A319445(n).
a(n) is the smallest e such that for any Eisenstein integer z coprime to n we have z^e == 1 (mod n).
By definition, A319445(n)/a(n) is always an integer, and is 1 iff (Z[w]/nZ[w])* is cyclic, that is, rank((Z[w]/nZ[w])*) = A319447(n) = 0 or 1, and n has a primitive root in (Z[w]/nZ[w])*. A319445(n)/a(n) = 1 iff n = 1, 3 or a prime congruent to 2 mod 3.
For n > 2, a(n) is divisible by 6.

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
Let G = (Z[w]/4Z[w])* = {1, w, 1 + w, w', 1 + w', -1 + 2w, -1, -w, -1 - w, -w', -1 - w', -1 + 2w'}. The possibilities for the exponent of G are 12, 6, 4, 3, 2 and 1. G^6 = {x^6 mod 4 : x belongs to G} = {1} and w^3 !== 1 (mod 4), w^4 !== 1 (mod 4). Therefore, the exponent of G is greater than 4, accordingly the exponent of G is 6 and a(4) = 6.

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

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), this sequence ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).

Equivalent in the ring of Gaussian integers: A227334.

f[p_, e_] := If[p == 3 , If[e == 1, 6, 2*3^(e - 1)], Switch[Mod[p, 3], 1, (p - 1)*p^(e - 1), 2, (p^2 - 1)*p^(e - 1)]]; eisPsi[1] = 1; eisPsi[n_] := LCM @@ f @@@ FactorInteger[n]; Array[eisPsi, 100]  (* Amiram Eldar, Feb 10 2020 *)

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==3, r=lcm(r,2*3^max(e-1,1)));
        if(p%3==1, r=lcm(r,(p-1)*p^(e-1)));
        if(p%3==2, r=lcm(r,(p^2-1)*p^(e-1)));
    );
    return(r);
}

a(3) = 6, a(3^e) = 2*3^(e-1) for e >= 2; a(p^e) = (p - 1)*p^(e-1) if p == 1 (mod 3) and (p^2 - 1)*p^(e-1) if p == 2 (mod 3). If gcd(m, n) = 1 then a(mn) = lcm(a(m), a(n)). [See the group structure of (Z[w]/(pi^e)Z[w])* in A319447, where pi is a prime element in Z[w]. - Jianing Song, Oct 03 2022]

Corrected by Jianing Song, Jan 12 2019

A319443 Number of distinct Eisenstein primes in the factorization of n.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Sep 19 2018

nonn

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).

			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.

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

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.

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 *)

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

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

A319444 Total number of factors in a factorization of n into Eisenstein primes.

Original entry on oeis.org

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

Jianing Song, Sep 19 2018

nonn

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).

			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.

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

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.

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 *)

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

Completely additive with a(p) = 2 if p = 3 or p == 1 (mod 3) and a(p) = 1 if p == 2 (mod 3).

A319448 Moebius function mu(n) defined for the Eisenstein integers.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0

Offset: 1

Jianing Song, Sep 19 2018

Just like the original Moebius function over the integers, a(n) = 0 if n has a squared Eisenstein prime factor, otherwise (-1)^t if n is a product of an Eisenstein unit and t distinct Eisenstein prime factors.

Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2. a(n) = 0 for n divisible by 3 since 3 = w'*(1 + w)^2 contains a squared factor. For rational primes p == 1 (mod 3), p is always factored as (x + y*w)(x + y*w'), x + y*w and x + y*w' are not associated so a(p) = (-1)*(-1) = 1.

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
a(14) = -1 because 14 is factored as 2*(2 + w)*(2 + w') with three distinct Eisenstein prime factors.
a(55) = (-1)*(-1) = 1 because 55 = 5*11 where 5 and 11 are congruent to 2 mod 3 (thus being Eisenstein primes).

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

Cf. A102283.
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), A319444 ("Omega", A001222), this sequence ("mu", A008683).
Equivalent in the ring of Gaussian integers: A318608.

f[p_, e_] := If[p == 3 || e > 1, 0, Switch[Mod[p, 3], 1, 1, 2, -1]]; eisMu[1] = 1; eisMu[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisMu, 100] (* Amiram Eldar, Feb 10 2020 *)

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==3||e>=2, r=0);
        if(Mod(p, 3)==2&e==1, r*=-1);
    );
    return(r);
}

a(n) = 0 if n is divisible by 3 or has a square prime factor, otherwise Product_{p divides n} (3 - 2*(p mod 3)) where the product is taken over the primes.
Multiplicative with a(p^e) = 0 if p = 3 or e > 1, a(p) = 1 if p == 1 (mod 3) and -1 if p == 2 (mod 3).
For squarefree n, a(n) = Legendre symbol (n, 3) = Kronecker symbol (-3, n) = A102283(n).

cp's OEIS Frontend

A319442 Number of divisors of n over the Eisenstein integers.

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A319445 Number of Eisenstein integers in a reduced system modulo n.

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A317797 Sum of the norm of divisors of n over Gaussian integers, with associated divisors counted only once.

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A319446 Exponent of the group of the Eisenstein integers in a reduced system modulo n.

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A319443 Number of distinct Eisenstein primes in the factorization of n.

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A319444 Total number of factors in a factorization of n into Eisenstein primes.

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A319448 Moebius function mu(n) defined for the Eisenstein integers.

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