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A152253 Array, by antidiagonals, arising in asymptotic approximation to the number of p-groups of order p^n. 0
1, 1, 1, 1, 1, 4, 1, 2, 9, 26, 1, 3, 25, 182, 612, 1, 4, 49, 2058, 26169, 65536, 1, 4, 121, 10148, 2964452, 43046721, 44503251, 1, 5, 169, 86491, 66831598, 152587890625, 1325604901966, 261120709453, 1, 5, 289, 190948, 4390610003, 33232930569601 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Poonen's abstract: The moduli space of rank-n commutative algebras equipped with an ordered basis is an affine scheme B_n of finite type over Z, with geometrically connected fibers. It is smooth if and only if n <= 3. It is reducible if n >= 8 (and the converse holds, at least if we remove the fibers above 2 and 3).

The relative dimension of B_n is (2/27) n^3 + O(n^{8/3}). The subscheme parameterizing etale algebras is isomorphic to GL_n/S_n, which is of dimension only n^2. For n >= 8, there exist algebras that are not limits of etale algebras. The dimension calculations lead also to asymptotic formulas for the number of commutative rings of order p^n and the dimension of the Hilbert scheme of n points in d-space for d >= n/2.

LINKS

Table of n, a(n) for n=1..42.

Bjorn Poonen, The moduli space of commutative algebras of finite rank, version 2, Mar 21, 2007.

FORMULA

Array, by antidiagonals, A[k,n] = floor(prime(k)^((2/27)*(n^3))), where prime(k) = A000040(k).

EXAMPLE

Array begins:

=========================================================================

k.|.p_k.|.n=1.|.n=2.|.n=3.|.....n=4.|......n=5.|..........n=6.|.....n=7.|

=========================================================================

1.|.2...|...1.|...1.|...4.|......26.|......612.|........65536.|.4403251.|

2.|.3...|...1.|...1.|...9.|.....182.|....26169.|......4304672.|.........|

3.|.5...|...1.|...2.|..25.|....2058.|..2964452.|.152587890625.|.........|

4.|.7...|...1.|...3.|..49.|...10148.|.66831599.|..............|.........|

5.|.11..|...1.|...4.|.121.|...86491.|..........|..............|.........|

6.|.13..|...1.|...4.|.169.|..190948.|..........|..............|.........|

7.|.17..|...1.|...5.|.289.|..681144.|..........|..............|.........|

8.|.19..|...1.|...5.|.361.|.1154088.|..........|..............|.........|

9.|.23..|...1.|...6.|.529.|.2854950.|..........|..............|.........|

10|.29..|...1.|...7.|.841.|.8567414.|..........|..............|.........|

MAPLE

Digits := 100: for d from 1 to 10 do for n from 1 to d do k := d-n+1 ; A := floor(ithprime(k)^(2*n^3/27)) ; printf("%d, ", A) ; od: od: # R. J. Mathar, Jan 22 2009

CROSSREFS

Cf. A000040, A000961.

Sequence in context: A264922 A233528 A084604 * A280440 A024569 A206692

Adjacent sequences:  A152250 A152251 A152252 * A152254 A152255 A152256

KEYWORD

easy,nonn,tabl

AUTHOR

Jonathan Vos Post, Nov 30 2008

EXTENSIONS

Corrected typo in A[4,3], reduced A[4,5] by 1, extended R. J. Mathar, Jan 22 2009

STATUS

approved

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Last modified May 21 00:41 EDT 2019. Contains 323427 sequences. (Running on oeis4.)