A319442 -id:A319442 - cp's OEIS frontend

A062327 Number of divisors of n over the Gaussian integers.

1, 3, 2, 5, 4, 6, 2, 7, 3, 12, 2, 10, 4, 6, 8, 9, 4, 9, 2, 20, 4, 6, 2, 14, 9, 12, 4, 10, 4, 24, 2, 11, 4, 12, 8, 15, 4, 6, 8, 28, 4, 12, 2, 10, 12, 6, 2, 18, 3, 27, 8, 20, 4, 12, 8, 14, 4, 12, 2, 40, 4, 6, 6, 13, 16, 12, 2, 20, 4, 24, 2, 21, 4, 12, 18, 10, 4, 24, 2, 36, 5, 12, 2, 20, 16, 6

Offset: 1

Reiner Martin, Jul 12 2001

Divisors which are associates are identified (two Gaussian integers z1, z2 are associates if z1 = u * z2 where u is a unit, i.e., one of 1, i, -1, -i).
a(A004614(n)) = A000005(n). - Vladeta Jovovic, Jan 23 2003
a(A004613(n)) = A000005(n)^2. - Benedikt Otten, May 22 2013

			For example, 5 has divisors 1, 1+2i, 2+i and 5.

Cf. A027748, A124010.
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): this sequence ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319442.

a062327 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f 2 e                  = 2 * e + 1
   f p e | p `mod` 4 == 1 = (e + 1) ^ 2
         | otherwise      = e + 1
-- Reinhard Zumkeller, Oct 18 2011

a:= n-> mul(`if`(i[1]=2, 2*i[2]+1, `if`(irem(i[1], 4)=3,
                 i[2]+1, (i[2]+1)^2)), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 09 2021

Table[Length[Divisors[n, GaussianIntegers -> True]], {n, 30}] (* Alonso del Arte, Jan 25 2011 *)
DivisorSigma[0,Range[90],GaussianIntegers->True] (* Harvey P. Dale, Mar 19 2017 *)

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*e+1));
        if(p%4==1,r*=(e+1)^2);
        if(p%4==3,r*=(e+1));
    );
    return(r);
}  \\ Joerg Arndt, Dec 09 2016

Presumably a(n) = 2 iff n is a rational prime == 3 mod 4 (see A045326). - N. J. A. Sloane, Jan 07 2003, Feb 23 2007

Multiplicative with a(2^e) = 2*e+1, a(p^e) = e+1 if p mod 4=3 and a(p^e) = (e+1)^2 if p mod 4=1. - Vladeta Jovovic, Jan 23 2003

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

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).

A319449 Sum of the norm of divisors of n over Eisenstein integers, with associated divisors counted only once.

Original entry on oeis.org

1, 5, 13, 21, 26, 65, 64, 85, 121, 130, 122, 273, 196, 320, 338, 341, 290, 605, 400, 546, 832, 610, 530, 1105, 651, 980, 1093, 1344, 842, 1690, 1024, 1365, 1586, 1450, 1664, 2541, 1444, 2000, 2548, 2210, 1682, 4160, 1936, 2562, 3146, 2650, 2210, 4433, 3249, 3255

Offset: 1

Jianing Song, Sep 19 2018

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

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2, and ||d|| denote the norm of d.
a(3) = ||1|| + ||1 + w|| + ||3|| = 1 + 3 + 9 = 13.
a(7) = ||1|| + ||2 + w|| + ||2 + w'|| + ||7|| = 1 + 7 + 7 + 49 = 64.

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

Cf. A001157.
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), this sequence ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).
Equivalent in the ring of Gaussian integers: A317797.

f[p_, e_] := If[p == 3 , DivisorSigma[1, 3^(2*e)], Switch[Mod[p, 3], 1, DivisorSigma[1, p^e]^2, 2, DivisorSigma[2, p^e]]]; eisSigma[1] = 1; eisSigma[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisSigma, 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*=((3^(2*e+1)-1)/2));
        if(Mod(p, 3)==1, r*=((p^(e+1)-1)/(p-1))^2);
        if(Mod(p, 3)==2, r*=(p^(2*e+2)-1)/(p^2-1));
    );
    return(r);
}

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

A332386 Numbers k such that k and k + 1 have the same number of divisors in Eisenstein integers.

Original entry on oeis.org

3, 7, 32, 50, 68, 174, 184, 200, 212, 219, 247, 291, 328, 343, 368, 376, 435, 472, 495, 543, 579, 608, 644, 679, 712, 716, 723, 788, 795, 849, 860, 871, 874, 904, 932, 939, 1011, 1015, 1058, 1074, 1076, 1159, 1184, 1220, 1227, 1336, 1359, 1384, 1436, 1495, 1515

Offset: 1

Amiram Eldar, Feb 10 2020

nonn

			3 is a term since 3 and 4 both have 3 divisors in Eisenstein integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005237, A319442, A332312, A332387, A332388.

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]; SequencePosition[eisNumDiv /@ Range[1520], {x_, x_}][[All, 1]] (* after Harvey P. Dale at A005237 *)

A332387 Numbers k such that k, k + 1 and k + 2 have the same number of divisors in Eisenstein integers.

Original entry on oeis.org

13448, 27848, 75774, 135400, 243338, 276123, 396950, 452823, 497575, 524823, 565674, 587575, 632224, 639848, 719223, 769316, 861123, 935799, 1060904, 1073875, 1153023, 1204312, 1308856, 1366624, 1413498, 1490599, 1555975, 1565223, 1601798, 1767424, 1902774, 1923295

Offset: 1

Amiram Eldar, Feb 10 2020

nonn

			13448 is a term since 13448, 13449 and 13450 each have 12 divisors in Eisenstein integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005238, A319442, A332313, A332386, A332388.

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]; Flatten[Position[Partition[ eisNumDiv /@ Range[10^6], 3, 1], {x_, x_, x_}]] (* after Harvey P. Dale at A005238 *)

A332388 Numbers k such that k, k + 1, k + 2 and k + 3 have the same number of divisors in Eisenstein integers.

Original entry on oeis.org

34193750, 76788050, 78267398, 113004199, 135383873, 148843670, 170293249, 199259222, 311313398, 318128599, 364828550, 368222599, 381026822, 384839047, 420686749, 428129222, 430154150, 432466824, 450050450, 462825847, 492828521, 510703975, 517126773, 518268772

Offset: 1

Amiram Eldar, Feb 10 2020

nonn

			34193750 is a term since 34193750, 34193751, 34193752 and 34193750 each have 24 divisors in Eisenstein integers.

Cf. A006601, A319442, A332314, A332386, A332387.

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]; m = 4; s = eisNumDiv /@ Range[m]; seq = {}; n = m + 1; While[Length[seq] < 10, If[Length @ Union[s] == 1, AppendTo[seq, n - m + 1]]; n++; s = Join[Rest[s], {eisNumDiv[n]}]]; seq

cp's OEIS Frontend

A062327 Number of divisors of n over the Gaussian integers.

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

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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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A319449 Sum of the norm of divisors of n over Eisenstein integers, with associated divisors counted only once.

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