A361871 The smallest order of a non-abelian group with an element of order n.

6, 6, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126

Offset

1,1

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Wikipedia, Lagrange's theorem (group theory).

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = 2*n for n >= 3.

Proof: By Lagrange's theorem in group theory we have that n divides a(n) for all n. A group of order n and with an element of order n is the cyclic group of order n, hence being abelian. On the other hand, the dihedral group D_{2n} is non-abelian for n >= 3 and contains an element of order n. - Jianing Song, Aug 11 2023

G.f.: 2*x*(3 - 3*x + x^3)/(1 - x)^2. - Andrew Howroyd, Nov 14 2025

From Elmo R. Oliveira, May 29 2026: (Start)

E.g.f.: x*(2*exp(x) + x + 4).

For n > 2: a(n) = 2*A000027(n) = A005843(n).

a(n) = 2*a(n-1) - a(n-2) for n > 4. (End)

Mathematica

Join[{6, 6}, 2*Range[3, 60]] (* Paolo Xausa, Jan 23 2026 *)

Crossrefs

Essentially the same as A163300, A103517, A051755.

Cf. A000027, A005843.

Sequence in context: A135357 A322346 A332559 * A179415 A179409 A186983

Adjacent sequences: A361868 A361869 A361870 * A361872 A361873 A361874

Keyword

nonn,easy

Author

Yue Yu, Apr 01 2023

Status

approved