cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Wouter Meeussen, Aug 25 2003

Keywords

Comments

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)}.

Examples

			{{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}}.

Crossrefs

Cf. A081959 A002464 A020063 A033169 A090410 A066052.

Programs

  • Mathematica
    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}]]

Extensions

a(6) from Wouter Meeussen, Nov 15 2005

A212818 Numbers up to 10^n with an even number of not necessarily distinct prime factors, or positive Liouville function.

Original entry on oeis.org

1, 5, 49, 493, 4953, 49856, 499735, 4999579, 49998058, 499987392, 4999941987, 49999828888, 499999738687, 4999999516711
Offset: 0

Views

Author

Martin Renner, May 28 2012

Keywords

Examples

			a(1) = 5 since up to 10 there are the five numbers 1, 4, 6, 9, 10 with an even number of prime factors, or positive Liouville function.

Links

Crossrefs

Cf. A055037 (goes up to n rather than 10^n), A002819, A008836, A028260, A065043, A090410.

Programs

  • Maple
    zg:=0: zu:=0: G:=[]: U:=[]: k:=0:
for i from 1 to 10^8 do if numtheory[bigomega](i) mod 2 = 0 then zg:=zg+1: else zu:=zu+1: fi: if i=10^k then G:=[op(G),zg]: U:=[op(U),zu]: k:=k+1: fi: od:
print(G);
  • Mathematica
    Table[Length[Select[Range[10^n], EvenQ[PrimeOmega[#]] &]], {n, 0, 5}] (* Alonso del Arte, May 28 2012 *)
Table[Count[LiouvilleLambda[Range[10^n]], 1], {n, 0, 5}] (* Ray Chandler, May 30 2012 *)

Formula

a(n) = A011557(n) - A212819(n).
a(n) = (10^n)/2 + A090410(n)/2. - Donovan Johnson, May 30 2012
a(n) = A055037(10^n). - Ray Chandler, May 30 2012

Extensions

a(9)-a(13) from Donovan Johnson, May 30 2012

A212819 Numbers up to 10^n with an odd number of prime factors, or negative Liouville function.

Original entry on oeis.org

0, 5, 51, 507, 5047, 50144, 500265, 5000421, 50001942, 500012608, 5000058013, 50000171112, 500000261313, 5000000483289
Offset: 0

Views

Author

Martin Renner, May 28 2012

Keywords

Examples

			a(1) = 5 since up to 10 there are the five numbers 2, 3, 5, 7, 8 with an odd number of prime factors or negative Liouville function.

Links

Crossrefs

Cf. A002819, A008836, A026424, A066829, A090410.

Programs

  • Maple
    zg:=0: zu:=0: G:=[]: U:=[]: k:=0:
for i from 1 to 10^8 do if numtheory[bigomega](i) mod 2 = 0 then zg:=zg+1: else zu:=zu+1: fi: if i=10^k then G:=[op(G),zg]: U:=[op(U),zu]: k:=k+1: fi: od:
print(U);
  • Mathematica
    Table[Count[LiouvilleLambda[Range[10^n]], -1], {n, 0, 5}] (* Ray Chandler, May 30 2012 *)

Formula

a(n) = A011557(n) - A212818(n).
a(n) = (10^n)/2 - A090410(n)/2. - Donovan Johnson, May 30 2012
a(n) = A055038(10^n). - Ray Chandler, May 30 2012

Extensions

a(9)-a(13) from Donovan Johnson, May 30 2012
Showing 1-3 of 3 results.

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