A346019
Number of n X n invertible matrices over GF(2) that have order 2^n-1.
Original entry on oeis.org
1, 2, 48, 2688, 1935360, 1919877120, 23222833643520, 335564785519165440, 65717007596073359769600, 21492090164219831579049984000, 66041307304745851496871108594892800, 226523509196861965428709270554756199219200, 16622838761287803491875715175557341313583022080000
Offset: 1
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a:= n-> mul(2^n-2^i, i=0..n-1)*numtheory[phi](2^n-1)/((2^n-1)*n):
seq(a(n), n=1..14); # Alois P. Heinz, Jul 01 2021
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nn = 13; Table[EulerPhi[2^n - 1]/n, {n, 1, nn}]* Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 1, nn}]/Table[2^n - 1, {n, 1, nn}]
A344873
Irregular triangle read by rows. T(n,k) is the number of n X n matrices over GF(2) whose characteristic polynomial is a product of k distinct squarefree irreducible factors.
Original entry on oeis.org
1, 0, 2, 0, 2, 6, 0, 48, 112, 0, 4032, 11520, 6720, 0, 1935360, 4952064, 2856960, 0, 2879815680, 9558687744, 7871496192, 0, 23222833643520, 66748107718656, 60247322394624, 15604761231360, 0, 629183972848435200, 2137709262359494656, 2101670528396820480, 465681743169454080
Offset: 0
Triangle begins:
1;
0, 2;
0, 2, 6;
0, 48, 112;
0, 4032, 11520, 6720;
0, 1935360, 4952064, 2856960;
0, 2879815680, 9558687744, 7871496192;
0, 23222833643520, 66748107718656, 60247322394624, 15604761231360;
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nn = 8; A001037 = Table[1/n Sum[MoebiusMu[n/d] 2^d, {d, Divisors[n]}], {n, 1, nn}];Prepend[Drop[Map[Prepend[#, 0] &,Map[Select[#, # > 0 &] &,Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 0,nn}] CoefficientList[Series[Product[(1 + v u^i/(2^i - 1))^A001037[[i]], {i, 1, nn}], {u, 0, nn}], {u, v}]]], 1], {1}] // Grid
Showing 1-2 of 2 results.
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