A340514 a(n) is the minimal order of a group in which all groups of order n can be embedded.
1, 2, 3, 8, 5, 12, 7, 32, 27, 20, 11, 144, 13, 28, 15, 256, 17, 216, 19, 160, 63, 44, 23
Offset: 1
References
- Heffernan, Robert, Des MacHale, and Brendan McCann. "Cayley's Theorem Revisited: Embeddings of Small Finite Groups." Mathematics Magazine 91.2 (2018): 103-111.
Links
- Heffernan, Robert, Des MacHale, and Brendan McCann, Minimal embeddings of small finite groups, arXiv:1706.09286 [math.GR], Jun 28 2017.
- MathStackExchange, How powerful is Cayley's theorem?, Oct 07 2021.
Formula
From David A. Craven, Oct 07 2021: (Start)
a(p)=p, a(p^2)=p^3, a(p^3)=p^6 if p is odd, a(8)=32.
If p
Extensions
a(16)-a(23) from David A. Craven, Oct 07 2021
A340516 Let p_i (i=1..m) denote the primes <= n, and let e_i be the maximum number such that p_i^e_i <= n; then a(n) = Product_{i=1..m} p_i^(2*e_i-1).
1, 2, 6, 24, 120, 120, 840, 3360, 30240, 30240, 332640, 332640, 4324320, 4324320, 4324320, 17297280, 294053760, 294053760, 5587021440, 5587021440, 5587021440, 5587021440, 128501493120, 128501493120, 3212537328000, 3212537328000, 28912835952000, 28912835952000, 838472242608000
Offset: 1
Keywords
References
- Heffernan, Robert, Des MacHale, and Brendan McCann. "Cayley’s Theorem Revisited: Embeddings of Small Finite Groups." Mathematics Magazine 91.2 (2018): 103-111.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..2242
- Heffernan, Robert, Des MacHale, and Brendan McCann, Minimal embeddings of small finite groups, arXiv:1706.09286 [math.GR], 2017. See Lemma 2.
Programs
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Mathematica
{1}~Join~Table[Times @@ Map[#^(2 Floor@ Log[#, n] - 1) &, Prime@ Range@ PrimePi@ n], {n, 2, 30}] (* Michael De Vlieger, Feb 23 2022 *)
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