A081959 -id:A081959 - cp's OEIS frontend

A081957 Triangle whose n-th row contains the n smallest numbers that are products of n distinct integers > 1, read by rows.

2, 6, 8, 24, 30, 36, 120, 144, 168, 180, 720, 840, 960, 1008, 1080, 5040, 5760, 6480, 6720, 7200, 7560, 40320, 45360, 50400, 51840, 55440, 57600, 60480, 362880, 403200, 443520, 453600, 483840, 498960, 518400, 524160, 3628800, 3991680, 4354560, 4435200, 4717440, 4838400, 4989600, 5080320, 5241600

Offset: 1

Amarnath Murthy, Apr 02 2003

			Triangle begins:
    2;
    6,   8;
   24,  30,  36;
  120, 144, 168, 180;
  ...

James Rayman, Rows n = 1..100, flattened
James Rayman, Python program

Cf. A081958, A081959, A081960.

Python
```
# See Rayman link.
```

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 08 2003
Terms a(39) and beyond from James Rayman, Jan 17 2021
Name edited by Michel Marcus, Jan 18 2021

A081958 Diagonal of A081957.

Original entry on oeis.org

2, 8, 36, 180, 1080, 7560, 60480, 524160, 5241600, 56609280, 670602240, 8622028800, 118879488000, 1793381990400, 28158588057600, 474249904128000, 8447576417280000, 159659194286592000, 3162772610629632000, 66117689869271040000, 1445143792856924160000

Offset: 1

Amarnath Murthy, Apr 02 2003

nonn

James Rayman, Table of n, a(n) for n = 1..400

Cf. A081957, A081959, A081960.

def a(n): return row(n)[-1] # row(n) is given in A081957. - James Rayman, Jan 17 2021

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 08 2003
a(11)-a(19) from James Rayman, Jan 16 2021
Corrected and extended by James Rayman, Jan 17 2021

A081960 Row products of A081957.

Original entry on oeis.org

2, 48, 25920, 522547200, 632073093120000, 68810005209415680000000, 922882872556916481982464000000000, 1930928443374224487368323197252403200000000000, 848408612299117898787190662782756632189835673600000000000000

Offset: 1

Amarnath Murthy, Apr 02 2003

nonn

James Rayman, Table of n, a(n) for n = 1..30

Cf. A081957, A081958, A081959.

from functools import reduce
import operator
def a(n): return reduce(operator.mul, row(n), 1) # row(n) is given in A081957. - James Rayman, Jan 18 2021

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 08 2003

More terms from James Rayman, Jan 18 2021

A072148 Number of invertible (-1,0,1) n X n matrices having (Tij = -Tji; i

Original entry on oeis.org

2, 14, 92, 796, 7672, 83944

Offset: 1

Wouter Meeussen, Aug 25 2003

The matrix powers T^k reach identity I for k a divisor of 12. All T^k are invertible (-1,0,1)-matrices with determinant +/-1. The matrix |Tij| is symmetric. The matrices T are "pseudo-anti-symmetric" (that is Tij=-Tji except for the main diagonal, or, equivalently, the sum of an anti-symmetric and a diagonal matrix). Their eigenvalues belong to {-1, -I, I, 1, -(-1)^(1/3), (-1)^(1/3), -(-1)^(2/3), (-1)^(2/3)}.

			{{1,-1,0,0,0},{1,0,0,0,0},{0,0,0,-1,0},{0,0,1,1,0},{0,0,0,0,-1}}
qualifies since its powers are:
{{0,-1,0,0,0},{1,-1,0,0,0},{0,0,-1,-1,0},{0,0,1,0,0},{0,0,0,0,1}},
{{-1,0,0,0,0},{0,-1,0,0,0},{0,0,-1,0,0},{0,0,0,-1,0},{0,0,0,0,-1}},
{{-1,1,0,0,0},{-1,0,0,0,0},{0,0,0,1,0},{0,0,-1,-1,0},{0,0,0,0,1}},
{{0,1,0,0,0},{-1,1,0,0,0},{0,0,1,1,0},{0,0,-1,0,0},{0,0,0,0,-1}},
{{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0},{0,0,0,0,1}}.

Cf. A081959 A002464 A020063 A033169 A090410 A066052.

triamatsig[li_List] := Block[{len=Sqrt[8Length[li]+1]/2-1/2}, If[IntegerQ[len], (Part[li, # ]&/@ Table[If[j>i, j(j-1)/2+i, i(i-1)/2+j], {i, len}, {j, len}])Table[If[j>i, -1, 1], {i, len}, {j, len}], li]]; n=4; it=triamatsig/@(-1+IntegerDigits[Range[0, -1+3^(n(n+1)/2)], 3, n(n+1)/2]); result4=Cases[it, (q_?MatrixQ)/; Det[q]=!=0 && And@@ Table[Union[Flatten[{MatrixPower[q, k], {-1, 0, 1}}]]==={-1, 0, 1}, {k, 25}]]

a(6) from Wouter Meeussen, Nov 15 2005

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A081957 Triangle whose n-th row contains the n smallest numbers that are products of n distinct integers > 1, read by rows.

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A081958 Diagonal of A081957.

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A081960 Row products of A081957.

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A072148 Number of invertible (-1,0,1) n X n matrices having (Tij = -Tji; i

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