A160498 -id:A160498 - cp's OEIS frontend

A114643 Number of real primitive Dirichlet characters modulo n.

1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1

Offset: 1

Steven Finch, Feb 16 2006

a(n) = 1 if either n or -n is a fundamental discriminant (not both); a(n) = 2 if n and -n are fundamental discriminants; a(n) = 0 otherwise. Also, Sum_{k=1..n} a(k) is asymptotic to (6/Pi^2)*n.
From Jianing Song, Feb 27 2019: (Start)
If n is an odd squarefree number, then a(n) = 1, where the unique real primitive Dirichlet character modulo n is {Kronecker(n,k)} = {Jacobi(k,n)} if n == 1 (mod 4) and {Kronecker(-n,k)} = {Jacobi(k,n)} if n == 3 (mod 4).
If n = 4*m, m is an odd squarefree number, then a(n) is also 1, where the unique real primitive Dirichlet character modulo n is {Kronecker(-n,k)} if m == 1 (mod 4) and {Kronecker(n,k)} if m == 3 (mod 4).
If n is 8 times an odd squarefree number, then a(n) = 2, where the two real primitive Dirichlet characters modulo n are {Kronecker(n,k)} and {Kronecker(-n,k)}.
a(n) = 0 if n == 2 (mod 4), n is divisible by 16 or the square of an odd prime. (End)
Mobius transform of A060594. - Jianing Song, Mar 02 2019

			From _Jianing Song_, Feb 27 2019: (Start)
For n = 5, the only real primitive Dirichlet characters modulo n is {Kronecker(5,k)} = [0, 1, -1, -1, 1] = A080891, so a(5) = 1.
For n = 8, the real primitive Dirichlet characters modulo n are {Kronecker(8,k)} = [0, 1, 0, -1, 0, -1, 0, 1] = A091337 and [0, 1, 0, 1, 0, -1, 0, -1] = A188510, so a(8) = 2.
For n = 20, the only real primitive Dirichlet characters modulo n is {Kronecker(-20,k)} = [0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1] = A289741, so a(20) = 1. (End)

W. Ellison and F. Ellison, Prime Numbers, Wiley, 1985, pp. 224-226.

Robert Israel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Cubic and quartic characters [Broken link]
Steven R. Finch, Cubic and quartic characters.
Vaclav Kotesovec, Graph - the asymptotic ratio
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
I. J. Zucker and M. M. Robertson, Some properties of Dirichlet L-series, J. Phys. A 9 (1976) 1207-1214.

Cf. A003657, A003658.
Cf. A160498 (number of cubic primitive Dirichlet characters modulo n), A160499 (number of quartic primitive Dirichlet characters modulo n).
Cf. A060594 (number of solutions to x^2 == 1 (mod n)).

A114643 := proc(n)
    local a,pf,p,r;
    a := 1 ;
    for pf in ifactors(n)[2] do
        p := op(1,pf);
        r := op(2,pf);
        if p = 2 then
            if r =  1 then
                a := 0 ;
            elif r =  2 then
                ;
            elif r =  3 then
                a := a*2 ;
            elif r >=  4 then
                a := 0 ;
            end if;
        else
            if r =1 then
                ;
            else
                a := 0 ;
            end if;
        end if;
    end do:
    a ;
end proc:
seq(A114643(n),n=1..40) ; # R. J. Mathar, Mar 02 2015
# Alternative:
f:= proc(n) local r,v,F;
  v:= padic:-ordp(n,2);
  if v = 1 or v >= 4 then return 0
  elif v = 3 then r:= 2
  else r:= 1
  fi;
  if numtheory:-issqrfree(n/2^v) then r else 0 fi
end proc:
map(f, [$1..100]); # Robert Israel, Oct 08 2017

a[n_] := Sum[ MoebiusMu[n/d] * Sum[ If[ Mod[i^2 - 1, d] == 0, 1, 0], {i, 2, d}], {d, Divisors[n]}]; a[1] = 1; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 20 2013, after Steven Finch *)
f[2, e_] := Which[e == 1, 0, e == 2, 1, e == 3, 2, e >= 4, 0]; f[p_, e_] := If[e == 1, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^2-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

This sequence is multiplicative with a(2) = 0, a(4) = 1, a(8) = 2, a(2^r) = 0 for r > 3, a(p) = 1 for prime p > 2 and a(p^r) = 0 for r > 1. - Steven Finch, Mar 08 2006 (With correction by Jianing Song, Jun 28 2018)

Dirichlet g.f.: zeta(s)*(1 + 2^(-2s) + 2^(1-3s))/(zeta(2s)*(1 + 2^(-s))). - R. J. Mathar, Jul 03 2011

A343023 Number of cyclic cubic fields with discriminant n^2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Jianing Song, Apr 02 2021

Equivalently, number of cubic fields with discriminant n^2. That is to say, it makes no difference if the word "cyclic" is omitted from the title.
Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.
Each term is 0 or a power of 2.
The first occurrence of 2^t is 9*A121940(t) for t >= 1.

			a(7) = 1 since there is only 1 (cyclic) cubic field with discriminant 7^2 = 49 is Q[x]/(x^3 - x^2 + x + 1).
a(63) = 2 since there are 2 (cyclic) cubic fields with discriminant 63^2 = 3969: Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).
a(819) = 4 since there are 4 (cyclic) cubic fields with discriminant 819^2 = 670761: Q[x]/(x^3 - 273x - 91), Q[x]/(x^3 - 273x - 728), Q[x]/(x^3 - 273x - 1547) and Q[x]/(x^3 - 273x - 1729).
a(35) = 0 since it is not of form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3). Indeed, there are no (cyclic) cubic fields with discriminant 35^2 = 1225.

Jianing Song, Table of n, a(n) for n = 1..16000
LMFDB, Cubic fields
Wikipedia, Cubic field
Ka Lun Wong, Maximal Unramified Extensions of Cyclic Cubic Fields, (2011), Theses and Dissertations, 2781.

Cf. A160498, A121940, A343000 (discriminants of cyclic cubic fields), A343001 (indices of positive terms).

a(n) = if(n<=1, 0, my(L=factor(n), w=omega(n)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 2^(w-1))

a(n) = A160498(n)/2 for n > 1.

A160499 Number of quartic primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 1, 2, 0, 0, 1, 1, 3, 0, 3, 4, 3, 0, 1, 3, 1, 0, 1, 2, 0, 0, 0, 1, 3, 0, 1, 0, 1, 0, 3, 0, 3, 0, 3, 6, 3, 0, 1, 1, 0, 0, 1, 4, 0, 0, 3, 3, 3, 0, 3, 2, 1, 0, 1, 3, 3, 0, 0, 0, 9, 0, 1, 3, 1, 0, 1, 0, 3, 0, 0, 1, 1, 0, 1, 12, 0

Offset: 1

Steven Finch, May 15 2009

Also called biquadratic primitive Dirichlet characters.
Primitive Dirichlet characters of both order 2 & order 4 are included.
a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a fourth-power root of unity (1, i, -1 and -i). - Jianing Song, Feb 27 2019
Mobius transform of A073103. - Jianing Song, Mar 02 2019

			From _Jianing Song_, Mar 02 2019: (Start)
For n = 5, the 3 quartic primitive Dirichlet characters modulo n are [0, 1, -1, -1, 1], [0, 1, i, -i, -1] and [0, 1, -i, i, -1], so a(5) = 3.
For n = 16, the 4 quartic primitive Dirichlet characters modulo n are [0, 1, 0, i, 0, i, 0, 1, 0, -1, 0, -i, 0, -i, 0, -1], [0, 1, 0, -i, 0, -i, 0, 1, 0, -1, 0, i, 0, i, 0, -1], [0, 1, 0, i, 0, -i, 0, -1, 0, -1, 0, -i, 0, i, 0, 1] and [0, 1, 0, -i, 0, i, 0, -1, 0, -1, 0, i, 0, -i, 0, 1], so a(16) = 4. (End)

Jianing Song, Table of n, a(n) for n = 1..10000
Steven R. Finch, Cubic and quartic characters.
Steven R. Finch, Cubic and quartic characters.
Steven R. Finch, Quartic and Octic Characters Modulo n, arXiv:0907.4894 [math.NT], 2016.

Cf. A114643 (number of quadratic primitive Dirichlet characters modulo n), A160498 (number of cubic primitive Dirichlet characters modulo n).
Cf. A073103 (number of solutions to x^4 == 1 (mod n)).
Cf. A064533.

f[n_] := Sum[If[Mod[k^4 - 1, n] == 0, 1, 0], {k, 1, n}]; a[n_] := Sum[ MoebiusMu[n/d]*f[d], {d, Divisors[n]}]; Table[a[n], {n, 2, 81}] (* Jean-François Alcover, Jun 19 2013 *)
f[2, e_] := Which[e == 1, 0, e == 2, 1, e == 3, 2, e == 4, 4, e >= 5, 0]; f[p_, 1] := If[Mod[p, 4] == 1, 3, 1]; f[p_, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^4-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

Multiplicative with a(4) = 1, a(8) = 2, a(16) = 4, a(2^e) = 0 for e = 1 or e >= 5; for odd primes p, a(p) = 3 if p == 1 (mod 4) and 1 if p == 3 (mod 4), a(p^e) = 0 for e >= 2. - Jianing Song, Mar 02 2019

Sum_{k=1..n} a(k) ~ c * n * log(n), where c = (7/(16*Pi*K^2)) * Product_{primes p == 1 (mod 4)} (1 - (5*p-3)/(p^2*(p+1))) = 0.1908767211685284480112237..., and K is the Landau-Ramanujan constant (A064533). - Amiram Eldar, Sep 16 2020

a(1) = 1 prepended by Jianing Song, Feb 27 2019

A343000 Discriminants of cyclic cubic fields.

Original entry on oeis.org

49, 81, 169, 361, 961, 1369, 1849, 3721, 3969, 4489, 5329, 6241, 8281, 9409, 10609, 11881, 13689, 16129, 17689, 19321, 22801, 24649, 26569, 29241, 32761, 37249, 39601, 44521, 47089, 49729, 52441, 58081, 61009, 67081, 73441, 76729, 77841, 80089, 90601, 94249, 97969

Offset: 1

Jianing Song, Apr 02 2021

Square terms in A006832.
Numbers of the form k^2 where A160498(k) >= 2.
Each term k^2 is associated with A343003(k) cyclic cubic fields.
Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.

			49 = 7^2 is a term since it is the discriminant of the cyclic cubic field Q[x]/(x^3 - x^2 + x + 1).
81 = 9^2 is a term since it is the discriminant of the cyclic cubic field Q[x]/(x^3 - 3x - 1).

Jianing Song, Table of n, a(n) for n = 1..3200
LMFDB, Cubic fields
Wikipedia, Cubic field
Ka Lun Wong, Maximal Unramified Extensions of Cyclic Cubic Fields, (2011), Theses and Dissertations, 2781.

Discriminants and their square roots of cyclic cubic fields:
At least 1 associated cyclic cubic field: this sequence, A343001.
Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
At least 2 associated cyclic cubic fields: A343024, A343025.
Exactly 2 associated cyclic cubic fields: A343002, A343003.
Cf. A006832, A160498, A343023.

isA343000(n) = if(issquare(n)&&n>1, my(k=sqrtint(n), L=factor(k), w=omega(k)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 1)

a(n) = A343001(n)^2.

A343001 Square roots of discriminants of cyclic cubic fields.

Original entry on oeis.org

7, 9, 13, 19, 31, 37, 43, 61, 63, 67, 73, 79, 91, 97, 103, 109, 117, 127, 133, 139, 151, 157, 163, 171, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 279, 283, 301, 307, 313, 331, 333, 337, 349, 367, 373, 379, 387, 397, 403, 409, 421, 427

Offset: 1

Jianing Song, Apr 02 2021

Numbers k such that k^2 is in A006832.
Numbers k such that A160498(k) >= 2.
Each term k is associated with A343003(k) cyclic cubic fields.
Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.

			7 is a term since 7^2 = 49 is the discriminant of the cyclic cubic field Q[x]/(x^3 - x^2 - 2*x + 1).
9 is a term since 9^2 = 81 is the discriminant of the cyclic cubic field Q[x]/(x^3 - 3*x - 1).

Jianing Song, Table of n, a(n) for n = 1..3200
LMFDB, Cubic fields
Wikipedia, Cubic field
Ka Lun Wong, Maximal Unramified Extensions of Cyclic Cubic Fields, (2011), Theses and Dissertations, 2781.

Discriminants and their square roots of cyclic cubic fields:
At least 1 associated cyclic cubic field: A343000, this sequence.
Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
At least 2 associated cyclic cubic fields: A343024, A343025.
Exactly 2 associated cyclic cubic fields: A343002, A343003.
Cf. A006832, A160498, A343023.

isA343001(n) = my(L=factor(n), w=omega(n)); for(i=1, w, if(!((L[i,1]%3==1 && L[i,2]==1) || L[i,1]^L[i,2] == 9), return(0))); (n>1)

a(n) = sqrt(A343001(n)).

A343022 Discriminants with exactly 1 associated cyclic cubic field.

Original entry on oeis.org

49, 81, 169, 361, 961, 1369, 1849, 3721, 4489, 5329, 6241, 9409, 10609, 11881, 16129, 19321, 22801, 24649, 26569, 32761, 37249, 39601, 44521, 49729, 52441, 58081, 73441, 76729, 80089, 94249, 97969, 109561, 113569, 121801, 134689, 139129, 143641, 157609, 167281, 177241

Offset: 1

Jianing Song, Apr 02 2021

A cubic field is cyclic if and only if its discriminant is a square. Hence all terms are squares.
Numbers of the form k^2 where A160498(k) = 2.
Numbers of the form k^2 where k is in A002476 U {9}. That is to say, numbers of the form k^2 where k = 9 or is a prime congruent to 1 modulo 3.
In general, there are exactly 2^(t-1) (cyclic) cubic fields with discriminant k^2 if and only if k is of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), See A343000 for more detailed information.

			169 is a term since the one (and only one) cyclic cubic field with that discriminant is Q[x]/(x^3 - x^2 - 4x - 1).

Jianing Song, Table of n, a(n) for n = 1..10000
LMFDB, Cubic fields
Wikipedia, Cubic field

Discriminants and their square roots of cyclic cubic fields:
At least 1 associated cyclic cubic field: A343000, A343001.
Exactly 1 associated cyclic cubic field: this sequence, A002476 U {9}.
At least 2 associated cyclic cubic fields: A343024, A343025.
Exactly 2 associated cyclic cubic fields: A343002, A343003.
Cf. A006832, A160498, A343023.

isA343022(n) = if(issquare(n), my(k=sqrtint(n)); k==9 || (isprime(k) && k%3==1), 0)

a(n) = A002476(n-1)^2 for n >= 3.

A343002 Discriminants with exactly 2 associated cyclic cubic fields.

Original entry on oeis.org

3969, 8281, 13689, 17689, 29241, 47089, 61009, 67081, 77841, 90601, 110889, 149769, 162409, 182329, 219961, 231361, 261121, 301401, 305809, 312481, 346921, 363609, 431649, 461041, 494209, 505521, 519841, 582169, 628849, 667489, 758641, 762129, 790321, 859329, 900601, 946729, 962361

Offset: 1

Jianing Song, Apr 02 2021

A cubic field is cyclic if and only if its discriminant is a square. Hence all terms are squares.
Numbers of the form k^2 where A160498(k) = 4.
Numbers of the form k^2 where k is of the form (i) 9p, where p is a prime congruent to 1 modulo 3; (ii) pq, where p, q are distinct primes congruent to 1 modulo 3.
Products of two nonequal terms in A343022.
In general, there are exactly 2^(t-1) (cyclic) cubic fields with discriminant k^2 if and only if k is of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), See A343000 for more detailed information.

			3969 = 63^2 is a term since it is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).
8281 = 91^2 is a term since it is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - x^2 - 30x + 64) and Q[x]/(x^3 - x^2 - 30x - 27).

Jianing Song, Table of n, a(n) for n = 1..10000
LMFDB, Cubic fields
Wikipedia, Cubic field

Discriminants and their square roots of cyclic cubic fields:
At least 1 associated cyclic cubic field: A343000, A343001.
Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
At least 2 associated cyclic cubic fields: A343024, A343025.
Exactly 2 associated cyclic cubic fields: this sequence, A343003.
Cf. A006832, A160498, A343023.

isA343002(n) = if(omega(n)==2, if(n==3969, 1, my(L=factor(n)); L[2,1]%3==1 && L[2,2]==2 && ((L[1,1]%3==1 && L[1,2]==2) || L[1,1]^L[1,2] == 81)), 0)

a(n) = A343003(n)^2.

A343003 Numbers k such that there are exactly 2 cyclic cubic fields with discriminant k^2.

Original entry on oeis.org

63, 91, 117, 133, 171, 217, 247, 259, 279, 301, 333, 387, 403, 427, 469, 481, 511, 549, 553, 559, 589, 603, 657, 679, 703, 711, 721, 763, 793, 817, 871, 873, 889, 927, 949, 973, 981, 1027, 1057, 1099, 1141, 1143, 1147, 1159, 1251, 1261, 1267, 1273, 1333

Offset: 1

Jianing Song, Apr 02 2021

It makes no difference if the word "cyclic" is omitted from the title because a cubic field is cyclic if and only if its discriminant is a square.
Numbers k such that A160498(k) = 4.
Numbers of the form (i) 9p, where p is a prime congruent to 1 modulo 3; (ii) pq, where p, q are distinct primes congruent to 1 modulo 3.
In general, there are exactly 2^(t-1) (cyclic) cubic fields with discriminant k^2 if and only if k is of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), See A343000 for more detailed information.

			63 is a term since 63^2 = 3969 is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).
91 is a term since 91^2 = 8281 is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - x^2 - 30x + 64) and Q[x]/(x^3 - x^2 - 30x - 27).

Jianing Song, Table of n, a(n) for n = 1..10000
LMFDB, Cubic fields
Wikipedia, Cubic field

Discriminants and their square roots of cyclic cubic fields:
At least 1 associated cyclic cubic field: A343000, A343001.
Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
At least 2 associated cyclic cubic fields: A343024, A343025.
Exactly 2 associated cyclic cubic fields: A343002, this sequence.
Cf. A006832, A160498, A343023.

isA343003(n) = if(omega(n)==2, if(n==63, 1, my(L=factor(n)); L[2,1]%3==1 && L[2,2]==1 && ((L[1,1]%3==1 && L[1,2]==1) || L[1,1]^L[1,2] == 9)), 0)

a(n) = sqrt(A343002(n)).

A343024 Discriminants with at least 2 associated cyclic cubic fields.

Original entry on oeis.org

3969, 8281, 13689, 17689, 29241, 47089, 61009, 67081, 77841, 90601, 110889, 149769, 162409, 182329, 219961, 231361, 261121, 301401, 305809, 312481, 346921, 363609, 431649, 461041, 494209, 505521, 519841, 582169, 628849, 667489, 670761, 758641, 762129, 790321, 859329, 900601, 946729, 962361

Offset: 1

Jianing Song, Apr 02 2021

A cubic field is cyclic if and only if its discriminant is a square. Hence all terms are squares.
Numbers of the form k^2 where A160498(k) >= 4.
Terms in A343000 that are not 81 or a square of a prime.
Different from A343002 since a(31) = 819^2 = (7*9*13)^2.
In general, there are exactly 2^(t-1) (cyclic) cubic fields with discriminant k^2 if and only if k is of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), See A343000 for more detailed information.

			8281 = 91^2 is a term since it is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - x^2 - 30x + 64) and Q[x]/(x^3 - x^2 - 30x - 27).
670761 = 819^2 is a term since it is the discriminant of the 4 cyclic cubic fields Q[x]/(x^3 - 273x - 91), Q[x]/(x^3 - 273x - 728), Q[x]/(x^3 - 273x - 1547) and Q[x]/(x^3 - 273x - 1729).

Jianing Song, Table of n, a(n) for n = 1..1600
LMFDB, Cubic fields
Wikipedia, Cubic field

Discriminants and their square roots of cyclic cubic fields:
At least 1 associated cyclic cubic field: A343000, A343001.
Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
At least 2 associated cyclic cubic fields: this sequence, A343025.
Exactly 2 associated cyclic cubic fields: A343002, A343003.
Cf. A006832, A160498, A343023.

isA343024(n) = if(issquare(n), my(k=sqrtint(n), L=factor(k), w=omega(k)); if(w<2, return(0)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 1)

a(n) = A343025(n)^2.

A343025 Numbers k such that there are at least 2 cyclic cubic fields with discriminant k^2.

Original entry on oeis.org

63, 91, 117, 133, 171, 217, 247, 259, 279, 301, 333, 387, 403, 427, 469, 481, 511, 549, 553, 559, 589, 603, 657, 679, 703, 711, 721, 763, 793, 817, 819, 871, 873, 889, 927, 949, 973, 981, 1027, 1057, 1099, 1141, 1143, 1147, 1159, 1197, 1251, 1261, 1267

Offset: 1

Jianing Song, Apr 02 2021

It makes no difference if the word "cyclic" is omitted from the title because a cubic field is cyclic if and only if its discriminant is a square.
Numbers k such that A160498(k) >= 4.
Terms in A343001 that are not 9 or a prime.
Different from A343002 since a(31) = 819 = 7*9*13.
In general, there are exactly 2^(t-1) (cyclic) cubic fields with discriminant k^2 if and only if k is of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3); see A343000 for more detailed information.

			63 is a term since 63^2 = 3969 is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).
819 is a term since 819^2 = 670761 is the discriminant of the 4 cyclic cubic fields Q[x]/(x^3 - 273x - 91), Q[x]/(x^3 - 273x - 728), Q[x]/(x^3 - 273x - 1547) and Q[x]/(x^3 - 273x - 1729).

Jianing Song, Table of n, a(n) for n = 1..1600
LMFDB, Cubic fields
Wikipedia, Cubic field

Discriminants and their square roots of cyclic cubic fields:
At least 1 associated cyclic cubic field: A343000, A343001.
Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
At least 2 associated cyclic cubic fields: A343024, this sequence.
Exactly 2 associated cyclic cubic fields: A343002, A343003.
Cf. A006832, A160498, A343023.

isA343025(n) = my(L=factor(n), w=omega(n)); if(w<2, return(0)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 1

a(n) = sqrt(A343024(n)).

cp's OEIS Frontend

A114643 Number of real primitive Dirichlet characters modulo n.

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A343023 Number of cyclic cubic fields with discriminant n^2.

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A160499 Number of quartic primitive Dirichlet characters modulo n.

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A343000 Discriminants of cyclic cubic fields.

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A343001 Square roots of discriminants of cyclic cubic fields.

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A343022 Discriminants with exactly 1 associated cyclic cubic field.

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A343002 Discriminants with exactly 2 associated cyclic cubic fields.

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