A341443 a(n) is the number of subfields of the (2n)-th cyclotomic field Q(zeta_(2n)) = Q(exp(2*Pi*i/(2n))).

1, 2, 2, 5, 3, 5, 4, 8, 4, 8, 4, 16, 6, 10, 8, 11, 5, 10, 6, 27, 10, 10, 4, 27, 6, 16, 6, 32, 6, 27, 8, 14, 10, 14, 16, 32, 9, 15, 16, 54, 8, 32, 8, 32, 16, 10, 4, 38, 8, 16, 14, 54, 6, 15, 16, 54, 15, 16, 4, 118, 12, 20, 30, 17, 30, 32, 8, 49, 10, 54, 8, 54, 12, 24, 16, 48, 20, 54, 8, 81

Offset

1,2

Comments

This is another version of A272831: here we only consider cyclotomic fields of even order since Q(zeta_n) = Q(zeta_(2n)) for odd n.

a(n) is the number of subgroups of the multiplicative group of integers modulo 2n.

Links

Table of n, a(n) for n=1..80.

ResearchGate, How many subfields does a cyclotomic field have

Formula

a(n) = A272831(2n).

Example

a(1) = 1: Q(zeta_2) = Q, which has itself as the only subfield.
a(2) = 2: Q(zeta_4) = Q(i) has 2 subfields: Q and Q(i).
a(3) = 2: Q(zeta_6) = Q(sqrt(-3)) has 2 subfields: Q and Q(sqrt(-3)).
a(4) = 5: Q(zeta_8) = Q(i, sqrt(2)) has 5 subfields: Q, Q(i), Q(sqrt(2)), Q(sqrt(-2)) and Q(zeta_8).
a(5) = 3: Q(zeta_10) = Q(zeta_5) has 3 subfields: Q, Q(sqrt(5)), Q(zeta_5).
a(6) = 5: Q(zeta_12) = Q(i, sqrt(-3)) has 5 subfields: Q, Q(i), Q(sqrt(-3)), Q(sqrt(3)) and Q(zeta_12).
a(7) = 4: Q(zeta_14) = Q(zeta_7) has 4 subfields: Q, Q(sqrt(-7)), Q((zeta_7)+(zeta_7)^(-1)) and Q(zeta_7).

Prog

(GAP) List([1..80], n->Sum( ConjugacyClassesSubgroups( LatticeSubgroups( GL(1, ZmodnZ(2*n)))), Size));

Crossrefs

Cf. A272831.

Sequence in context: A174608 A130327 A224361 * A224166 A286109 A239665

Adjacent sequences: A341440 A341441 A341442 * A341444 A341445 A341446

Keyword

nonn

Author

Jianing Song, Feb 12 2021

Status

approved