A104452 Number of groups of order <= n all of whose subgroups are normal.

1, 2, 3, 5, 6, 7, 8, 12, 14, 15, 16, 18, 19, 20, 21, 27, 28, 30, 31, 33, 34, 35, 36, 40, 42, 43, 46, 48, 49, 50, 51, 59, 60, 61, 62, 66, 67, 68, 69, 73, 74, 75, 76, 78, 80, 81, 82, 88, 90, 92, 93, 95, 96, 99, 100, 104, 105, 106, 107, 109, 110, 111, 113, 125, 126, 127

Offset

1,2

References

Robert D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.

John C. Lennox and Stewart. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, On the number of hamiltonian groups, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; arXiv preprint, arXiv:math/0503183 [math.CO], 2005.

Tomaž Pisanski and Thomas W. Tucker, The genus of low rank hamiltonian groups, Discrete Math. 78 (1989), 157-167.

Eric Weisstein's World of Mathematics, Abelian Group.

Eric Weisstein's World of Mathematics, Hamiltonian Group.

Formula

a(n) ~ c * n, where c = A021002 * (1 + A048651/4) = 2.46053840757488111675... . - Amiram Eldar, Oct 03 2023

Mathematica

orders[n_]:=Map[Last, FactorInteger[n]]; a[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/; Mod[n, 8]==0:=a[e[n]]; h[n_]:=0; numberOfAbelianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[a, Table[i, {i, 1, n}]], i], {i, 1, n}]]; numberOfHamiltonianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[h, Table[i, {i, 1, n}]], i], {i, 1, n}]]; numberOfAllGroupsOfOrderLEQThanN[n_]:=numberOfAbelianGroupsOfOrderLEQThanN[n] +numberOfHamiltonianGroupsOfOrderLEQThanN[n];

Crossrefs

Partial sums of A104404.

Cf. A000688, A021002, A048651, A063966, A104488, A104407, A104453.

Sequence in context: A352328 A028229 A394263 * A335073 A344514 A394264

Adjacent sequences: A104449 A104450 A104451 * A104453 A104454 A104455

Keyword

nonn,easy

Author

Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), Tomaz Pisanski, Apr 19 2005

Status

approved