cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

Eric W. Weisstein

Keywords

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

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

A131131 4*A007318 - 3*A097806.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 4, 12, 9, 1, 4, 16, 24, 13, 1, 4, 20, 40, 40, 17, 1, 4, 24, 60, 80, 60, 21, 1, 4, 28, 84, 140, 140, 84, 25, 1, 4, 32, 112, 224, 280, 224, 112, 29, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 16 2007

Keywords

Comments

Row sums = A131130, (1, 2, 10, 26, 52, 98, 190, ...), the binomial transform of (1, 1, 7, 1, 7, 1, ...). Generally, triangles generated from N*A007318 - (N-1)*A097806 have row sums that are binomial transforms of (1, 1, (N-1), 1, (N-1), 1, ...). A095121 = (1, 2, 6, 14, 30, 62, ...), the binomial transform of (1, 1, 3, 1, 3, 1, ...) and = row sums of A131108.
Triangle T(n,k), 0 <= k <= n,read by rows given by [1,3,-4,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

Examples

			First few rows of the triangle:
  1;
  1,  1;
  4,  5,  1;
  4, 12,  9,  1;
  4, 16, 24, 13,  1
  4, 20, 40, 40, 17,  1;
  ...

Crossrefs

Cf. A131130, A131129, A131128, A131127, A046055, A131108, A095121, A097806.

Formula

4*A007318 - 3*A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.
G.f.: (1-x*y+3*x^2+3*x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 12 2015
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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