cp's OEIS Frontend

Also called primitive Dirichlet characters of order 3.

Mobius transform of A060839.

C. David, J. Fearnley & H. Kisilevsky prove that Sum_{k=1..n} a(k) ~ C*n, with C = (11*sqrt(3)/(18*Pi)) * Product_{primes p == 1 (mod 3)} (1 - 2/(p*(p+1))) = 0.3170565167922841205670156...; they credit Cohen, F. Diaz y Diaz, & M. Olivier 2002 (see Proposition 5.2. and Corollary 5.3.). - Charles R Greathouse IV, Aug 26 2009 [corrected by Vaclav Kotesovec, Sep 16 2020]

a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a cubic root of unity: 1, w = (-1 + sqrt(3)*i)/2 or w^2 = (-1 - sqrt(3)*i)/2. - Jianing Song, Feb 27 2019

Every term is 0 or a power of 2. - Jianing Song, Mar 02 2019

From Jianing Song, Apr 03 2021: (Start)

For n >= 2, a(n) is the number of cyclic cubic fields with discriminant n^2. See A343023 for detailed information.

The first occurrence of 2^t is 9*A121940(t-1) for t >= 2. (End)

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.

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.

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.

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.

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.

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.

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.