|
%I #12 Aug 06 2017 21:11:22
%S 1,1,1,1,1,4,1,2,9,26,1,3,25,182,612,1,4,49,2058,26169,65536,1,4,121,
%T 10148,2964452,43046721,44503251,1,5,169,86491,66831598,152587890625,
%U 1325604901966,261120709453,1,5,289,190948,4390610003,33232930569601
%N Array, by antidiagonals, arising in asymptotic approximation to the number of p-groups of order p^n.
%C 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).
%C 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.
%H Bjorn Poonen, <a href="http://arxiv.org/abs/math/0608491">The moduli space of commutative algebras of finite rank</a>, version 2, Mar 21, 2007.
%F Array, by antidiagonals, A[k,n] = floor(prime(k)^((2/27)*(n^3))), where prime(k) = A000040(k).
%e Array begins:
%e =========================================================================
%e k.|.p_k.|.n=1.|.n=2.|.n=3.|.....n=4.|......n=5.|..........n=6.|.....n=7.|
%e =========================================================================
%e 1.|.2...|...1.|...1.|...4.|......26.|......612.|........65536.|.4403251.|
%e 2.|.3...|...1.|...1.|...9.|.....182.|....26169.|......4304672.|.........|
%e 3.|.5...|...1.|...2.|..25.|....2058.|..2964452.|.152587890625.|.........|
%e 4.|.7...|...1.|...3.|..49.|...10148.|.66831599.|..............|.........|
%e 5.|.11..|...1.|...4.|.121.|...86491.|..........|..............|.........|
%e 6.|.13..|...1.|...4.|.169.|..190948.|..........|..............|.........|
%e 7.|.17..|...1.|...5.|.289.|..681144.|..........|..............|.........|
%e 8.|.19..|...1.|...5.|.361.|.1154088.|..........|..............|.........|
%e 9.|.23..|...1.|...6.|.529.|.2854950.|..........|..............|.........|
%e 10|.29..|...1.|...7.|.841.|.8567414.|..........|..............|.........|
%p 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
%Y Cf. A000040, A000961.
%K easy,nonn,tabl
%O 1,6
%A _Jonathan Vos Post_, Nov 30 2008
%E Corrected typo in A[4,3], reduced A[4,5] by 1, extended _R. J. Mathar_, Jan 22 2009
|
