A046055 Orders of finite Abelian groups having the incrementally largest numbers of nonisomorphic forms (A046054).

1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 221184, 262144, 442368, 524288, 663552, 884736, 995328, 1048576, 1327104, 1769472, 1990656, 2097152, 2654208, 3538944, 3981312, 4194304

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• nice

Comments

Different from A151821, but often confused with it.

Nicolas used the notation a(n) for the number of Abelian groups of order n (A000688) and named these numbers a-highly composite numbers (a-hautement composés). - Amiram Eldar, Aug 20 2019

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1111 (terms 1..216 from Charlie Neder)
• H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
• Jean-Louis Nicolas, Sur les entiers N pour lesquels il y a beaucoup de groupes abéliens d’ordre N, Annales de l'institut Fourier. Vol. 28, No. 4. (1978), pp. 1-16, alternative link.
• Eric Weisstein's World of Mathematics, Abelian Group.
• Wikipedia, Abelian group
• Index entries for sequences related to groups

Crossrefs

Cf. A000079, A000688, A046054, A046056, A010684.

Warning: this is different from A151821.

Programs

Mathematica

aa = {}; max = 0; Do[If[FiniteAbelianGroupCount[n] > max, max = FiniteAbelianGroupCount[n]; AppendTo[aa, n]], {n, 2^22}]; aa (* Artur Jasinski, Oct 06 2011 *)

Formula

Warning: the g.f. is not x*(1+2*x)/(1-2*x), as claimed earlier.

Warning: this is not the binomial transform of A010684, as claimed earlier.

Warning: this is not the row sums of either A131127 or A134058, as claimed earlier.

Extensions

More terms from David Wasserman, Feb 06 2002

Many incorrect formulas and assertions deleted by R. J. Mathar, Jul 08 2009

Edited by N. J. A. Sloane, Jul 08 2009