A005848 -id:A005848 - cp's OEIS frontend

A330565 The thirteen entries from A005848 for which the integers of the cyclotomic field form a Euclidean ring with respect to the norm.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 12, 15

Offset: 1

N. J. A. Sloane, Jan 05 2020

Hendrik W. Lenstra and A. J. van der Poorten, Euclidean number fields 1, Math. Intelligencer 2 (1979): pp. 6-15. See Eq. (13).

Cf. A005848.

A061653 Relative class number h- of cyclotomic field Q(zeta_n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 8, 1, 9, 1, 1, 1, 1, 1, 37, 1, 2, 1, 121, 1, 211, 1, 1, 3, 695, 1, 43, 1, 5, 3, 4889, 1, 10, 2, 9, 8, 41241, 1, 76301, 9, 7, 17, 64, 1, 853513, 8, 69, 1, 3882809, 3, 11957417, 37, 11, 19, 1280, 2, 100146415

Offset: 1

N. J. A. Sloane, Jun 16 2001

Note that if n == 2 (mod 4), Q(zeta_n) is the same field as Q(zeta_{n/2}).
From Richard N. Smith, Jul 15 2019: (Start)
For prime p, p divides a(p) (or a(2p)) if and only if p is in A000928.
For prime p, p divides a(4p) if and only if p is in A250216. (End)

			Q(zeta_23) = 3 is the first time that h- is bigger than 1.

Richard N. Smith, Table of n, a(n) for n = 1..256 (terms 1..163 from R. J. Mathar)
Dylan Johnston, Diego Martín Duro, and Dmitriy Rumynin, Disconnected Reductive Groups: Classification and Representations, arXiv:2409.06375 [math.RT], 2024. See p. 14.
L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 353. [WARNING: The table contains errors for n=59, 97, ...]

Contains A000927, A035115, A061494 as subsequences.

Cf. A055513, A005848.

For prime p, a(p) = A000927(A000720(p)).

Washington gives an extensive table on pp. 353-360.
Missing term a(1) = 1 inserted by N. J. A. Sloane, Feb 05 2009 at the suggestion of Tanya Khovanova
More terms from R. J. Mathar, Feb 06 2009
a(59) changed from 41421 to 41241 (given correctly in 2nd edition of Washington), Matthew Johnson, Jul 20 2013
a(59) in b-file changed as above by Andrew Howroyd, Feb 23 2018
a(97) corrected, a(163) added by Max Alekseyev, Mar 05 2018

A193175 Numbers n such that there exists a two-dimensional generalized Bravais lattice with n-fold symmetry.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 48, 50, 54, 60, 66, 70, 84, 90

Offset: 1

N. J. A. Sloane, Jul 17 2011

The complete list of 29 terms is shown. These are the values of 2n such that A193179(n)=1.

Mermin, N. David; Rokhsar, Daniel S.; and Wright, David C.; Beware of 46-fold symmetry: the classification of two-dimensional quasicrystallographic lattices. Phys. Rev. Lett. 58 (1987), 2099-2101.

Cf. A193176, A193179, A005848.

A380784 Prime numbers p where the cyclotomic field Q(zeta_(p-1)) has class number one.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 61, 67, 71

Offset: 1

Steven Lu, Feb 02 2025

For a prime number p, the cyclotomic field of power p-1 can take a significant part in Z/pZ or p-adic field Q_p, since 1~p-1 are all (p-1)-th unit roots in Z/pZ. It would be much better if the cyclotomic integer ring is a unique factorization domain.

A prime number p is in this sequence if and only if (p-1)/2 is in A005848 (if p equals 3 modulus 4) or p-1 is in A005848 (otherwise).

Cf. A005848.

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A330565 The thirteen entries from A005848 for which the integers of the cyclotomic field form a Euclidean ring with respect to the norm.

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A061653 Relative class number h- of cyclotomic field Q(zeta_n).

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A193175 Numbers n such that there exists a two-dimensional generalized Bravais lattice with n-fold symmetry.

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A380784 Prime numbers p where the cyclotomic field Q(zeta_(p-1)) has class number one.

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