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A152253
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Array, by antidiagonals, arising in asymptotic approximation to the number of p-groups of order p^n.
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0
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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
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OFFSET
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1,6
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COMMENTS
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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.
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LINKS
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FORMULA
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Array, by antidiagonals, A[k,n] = floor(prime(k)^((2/27)*(n^3))), where prime(k) = A000040(k).
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EXAMPLE
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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.|..........|..............|.........|
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MAPLE
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Corrected typo in A[4,3], reduced A[4,5] by 1, extended R. J. Mathar, Jan 22 2009
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STATUS
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approved
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