A051731 -id:A051731 - cp's OEIS frontend

A160182 Triangle read by rows, 1 / ((-1)A129184 A051731 + I), I = Identity matrix.

1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 6, 2, 1, 1, 1, 1, 10, 4, 2, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 1, 16, 6, 3, 2, 1, 1, 1, 1, 1, 19, 7, 4, 2, 1, 1, 1, 1, 1, 1, 26, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, May 03 2009

Inverse mobius transform (A051731) * the triangle shifts row terms to the right deleting the right border, getting triangle A160183: (1; 2,1; 3,1,1; 5,2,1,1;...).

			First few rows of the triangle:
   1;
   1,  1;
   2,  1,  1;
   3,  1,  1,  1;
   5,  2,  1,  1,  1;
   6,  2,  1,  1,  1,  1;
  10,  4,  2,  1,  1,  1,  1;
  11,  4,  2,  1,  1,  1,  1,  1;
  16,  6,  3,  2,  1,  1,  1,  1,  1;
  19,  7,  4,  2,  1,  1,  1,  1,  1,  1;
  26, 10,  5,  3,  2,  1,  1,  1,  1,  1,  1;
  ...

Cf. A051731, A129184, A160183.

Row sums = A068336. Left border = A003238.

A160182den := proc(n,k)
    a := add( A129184(n,i)*A051731(i,k),i=1..n) ;
    if n =k then
        -a+1 ;
    else
        -a;
    end if;
end proc:
N := 20 :
M := Matrix(N,N) :
for n from 1 to N do
for k from 1 to N do
    M[n,k] := A160182den(n,k) ;
end do:
end do:
MatrixInverse(M) ;  # R. J. Mathar, Aug 04 2015

Triangle read by rows, 1 / ((-1)*A129184 * A051731 + I), I = Identity matrix. The operations shift the inverse Mobius transform (A051731) down, changing the signs to (-1), then add I = (1,1,1,...) as the right border.

A174852 Triangle T(n,k) read by rows. T(1,1)=1, n>1 T(n,1)=A049240, k>1 T(n,k)=A051731.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Mats Granvik, Mar 31 2010

Except for the first term the first column is equal to A049240. The rest of the table equals A051731. The first column = 0 when n is a square greater than 1.

			Table begins:
1,
1,1,
1,0,1,
0,1,0,1,
1,0,0,0,1,
1,1,1,0,0,1,
1,0,0,0,0,0,1,
1,1,0,1,0,0,0,1,
0,0,1,0,0,0,0,0,1,
1,1,0,0,1,0,0,0,0,1,

Cf. A174854.

A177975 Square array T(n,k) read by antidiagonals up. Each column is the first column in the matrix inverse of a triangular matrix that is the k-th differences of A051731 in the column direction.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 4, 7, 9, 4, 1, 0, 2, 14, 16, 14, 5, 1, 0, 6, 13, 34, 30, 20, 6, 1, 0, 4, 27, 43, 69, 50, 27, 7, 1, 0, 6, 26, 83, 107, 125, 77, 35, 8, 1, 0, 4, 39, 100, 209, 226, 209, 112, 44, 9, 1, 0, 10, 38, 155, 295, 461, 428, 329, 156, 54, 10, 1

Offset: 1

Mats Granvik, May 16 2010

The first column in this array is the first column in A134540 which is the matrix inverse of A177978. The second column is the first column in A159905. The rows in this array are described by both binomial expressions and closed form polynomials.

			Table begins:
  1..1...1...1....1.....1.....1......1......1.......1.......1
  0..1...2...3....4.....5.....6......7......8.......9......10
  0..2...5...9...14....20....27.....35.....44......54......65
  0..2...7..16...30....50....77....112....156.....210.....275
  0..4..14..34...69...125...209....329....494.....714....1000
  0..2..13..43..107...226...428....749...1234....1938....2927
  0..6..27..83..209...461...923...1715...3002....5004....8007
  0..4..26.100..295...736..1632...3312...6270...11220...19162
  0..6..39.155..480..1266..2975...6399..12825...24255...43692
  0..4..38.182..641..1871..4789..11103..23807...47896...91367
  0.10..65.285.1000..3002..8007..19447..43757...92377..184755
  0..4..50.292.1209..4066.11837..30920..74139..165748..349438
  0.12..90.454.1819..6187.18563..50387.125969..293929..646645
  0..6..75.473.2166..8101.26202..75797.200479..492406.1136048
  0..8.100.636.2976.11482.38523.115915.319231..816421.1960190
  0..8.100.696.3546.14712.52548.167112.483879.1296064.3249312

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Column k=1..5 gives A063524, A000010, A007438, A117108, A117109.
Main diagonal gives A332470.
Cf. A177976, A177977.

T(n, k) = sumdiv(n, d, moebius(n/d)*binomial(d+k-2, d-1)); \\ Seiichi Manyama, Jun 12 2021

From Seiichi Manyama, Jun 12 2021: (Start)
G.f. of column k: Sum_{j>=1} mu(j) * x^j/(1 - x^j)^k.
T(n,k) = Sum_{d|n} mu(n/d) * binomial(d+k-2,d-1). (End)

A177978 Triangle T(n,k) read by rows: A051731(n,k) - A051731(n-1,k).

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 1, -1, 1, 0, -1, 0, -1, 1, 0, 1, 1, 0, -1, 1, 0, -1, -1, 0, 0, -1, 1, 0, 1, 0, 1, 0, 0, -1, 1, 0, -1, 1, -1, 0, 0, 0, -1, 1, 0, 1, -1, 0, 1, 0, 0, 0, -1, 1, 0, -1, 0, 0, -1, 0, 0, 0, 0, -1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, -1, 1, 0, -1, -1, -1, 0, -1, 0, 0, 0, 0, 0, -1, 1, 0, 1

Offset: 1

Mats Granvik, May 16 2010

The recurrence for this triangle is similar to the recurrence in A177517. Cumulative column sums give table A051731.

			Triangle begins:
  1;
  0,  1;
  0, -1,  1;
  0,  1, -1,  1;
  0, -1,  0, -1,  1;
  0,  1,  1,  0, -1,  1;
  0, -1, -1,  0,  0, -1,  1;
  0,  1,  0,  1,  0,  0, -1,  1;
  0, -1,  1, -1,  0,  0,  0, -1,  1;
  0,  1, -1,  0,  1,  0,  0,  0, -1,  1;
  0, -1,  0,  0, -1,  0,  0,  0,  0, -1,  1;
  0,  1,  1,  1,  0,  1,  0,  0,  0,  0, -1,  1;
  0, -1, -1, -1,  0, -1,  0,  0,  0,  0,  0, -1,  1;
  0,  1,  0,  0,  0,  0,  1,  0,  0,  0,  0,  0, -1,  1;

Matrix inverse of A134540. Cf. A177517.

T(n,1)=A000007, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - (Sum_{i=1..k-1} T(n-i,k)).

Typo in sequence (erroneous comma) corrected by N. J. A. Sloane, May 22 2010

Edited by Mats Granvik, Dec 11 2010

A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

Original entry on oeis.org

1, 1, -1, -1, -1, 1, -1, 0, 2, -1, 0, 0, 2, -3, 1, -1, 2, 1, -5, 4, -1, 1, -3, 5, -8, 9, -5, 1, -1, 4, -4, -5, 15, -14, 6, -1, 0, -1, 6, -17, 29, -31, 20, -7, 1, 0, 0, 2, -13, 36, -55, 50, -27, 8, -1, 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1

Offset: 0

Mats Granvik, Jul 24 2016

From Mats Granvik, Sep 30 2017: (Start)
Conjecture: The largest absolute value of the eigenvalues of these characteristic polynomials appear to have the same prime signature in the factorization of the matrix sizes N.
In other words: Let b(N) equal the sequence of the largest absolute values of the eigenvalues of the characteristic polynomials of the matrices of size N. b(N) is then a sequence of truncated eigenvalues starting:
b(N=1..infinity)
= 1.00000, 1.61803, 1.61803, 2.00000, 1.61803, 2.20557, 1.61803, 2.32472, 2.00000, 2.20557, 1.61803, 2.67170, 1.61803, 2.20557, 2.20557, 2.61803, 1.61803, 2.67170, 1.61803, 2.67170, 2.20557, 2.20557, 1.61803, 3.08032, 2.00000, 2.20557, 2.32472, 2.67170, 1.61803, 2.93796, 1.61803, 2.89055, 2.20557, 2.20557, 2.20557, 3.21878, 1.61803, 2.20557, 2.20557, 3.08032, 1.61803, 2.93796, 1.61803, 2.67170, 2.67170, 2.20557, 1.61803, 3.45341, 2.00000, 2.67170, 2.20557, 2.67170, 1.61803, 3.08032, 2.20557, 3.08032, 2.20557, 2.20557, 1.61803, 3.53392, 1.61803, 2.20557, 2.67170, ...
It then appears that for n = 1,2,3,4,5,...,infinity we have the table:
Prime signature: b(Axxxxxx(n)) = Largest abs(eigenvalue):
p^0      : b(1)          = 1.0000000000000000000000000000...
p        : b(A000040(n)) = 1.6180339887498949025257388711...
p^2      : b(A001248(n)) = 2.0000000000000000000000000000...
p*q      : b(A006881(n)) = 2.2055694304005917238953315973...
p^3      : b(A030078(n)) = 2.3247179572447480566665944934...
p^2*q    : b(A054753(n)) = 2.6716998816571604358216518448...
p^4      : b(A030514(n)) = 2.6180339887498917939012699207...
p^3*q    : b(A065036(n)) = 3.0803227214906021558249449299...
p*q*r    : b(A007304(n)) = 2.9379558827528557962693867011...
p^5      : b(A050997(n)) = 2.8905508875432590620846440288...
p^2*q^2  : b(A085986(n)) = 3.2187765853016649941764626419...
p^4*q    : b(A178739(n)) = 3.4534111136673804054453285061...
p^2*q*r  : b(A085987(n)) = 3.5339198574905377192578725953...
p^6      : b(A030516(n)) = 3.1478990357047909043330946587...
p^3*q^2  : b(A143610(n)) = 3.7022736187975437971431347250...
p^5*q    : b(A178740(n)) = 3.8016448153137023524550386355...
p^3*q*r  : b(A189975(n)) = 4.0600260453688532535920785448...
p^7      : b(A092759(n)) = 3.3935083220984414431597997463...
p^4*q^2  : b(A189988(n)) = 4.1453038440113498808159420150...
p^2*q^2*r: b(A179643(n)) = 4.2413382309993874486053755390...
p^6*q    : b(A189987(n)) = 4.1311805192254587026923218218...
p*q*r*s  : b(A046386(n)) = 3.8825338629275134572083061357...
...
b(Axxxxxx(1)) in the sequences above, is given by A025487.
(End)
First column in the coefficients of the characteristic polynomials is the Möbius function A008683.
Row sums of coefficients start: 0, -1, 0, 0, 0, 0, 0, 0, 0, ...
Third diagonal is a signed version of A000096.
Most of the eigenvalues are equal to 1. The number of eigenvalues equal to 1 are given by A075795 for n>1.
The first three of the eigenvalues above can be calculated as nested radicals. The fourth eigenvalue 2.205569430400590... minus 1 = 1.205569430400590... is also a nested radical.

			{
{ 1},
{ 1, -1},
{-1, -1,  1},
{-1,  0,  2,  -1},
{ 0,  0,  2,  -3,  1},
{-1,  2,  1,  -5,  4,   -1},
{ 1, -3,  5,  -8,  9,   -5,   1},
{-1,  4, -4,  -5, 15,  -14,   6,  -1},
{ 0, -1,  6, -17, 29,  -31,  20,  -7,  1},
{ 0,  0,  2, -13, 36,  -55,  50, -27,  8, -1},
{ 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1}
}

Mats Granvik, Mathematica program to verify that eigenvalues determine prime signature
OEIS Wiki, Prime signatures
Eric Weisstein, Prime signature
Index to sequences related to prime signature

Cf. A051731, A008683, A000040, A001248, A006881, A030078, A030514, A054753, A000096, A001622, A272874, A075795.

Cf. A025487. - Mats Granvik, Sep 30 2017

Clear[x, AA, nn, s]; Monitor[AA = Flatten[Table[A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; MatrixForm[A]; a = A[[1, nn]]; A[[1, nn]] = A[[nn, nn]]; A[[nn, nn]] = a; CoefficientList[CharacteristicPolynomial[A, x], x], {nn, 1, 10}]], nn]

A127471 Triangle formed from the matrix product A051731 * A054522 of infinite lower triangular matrices, read by rows.

Original entry on oeis.org

1, 2, 1, 2, 0, 2, 3, 2, 0, 2, 2, 0, 0, 0, 4, 4, 2, 4, 0, 0, 2, 2, 0, 0, 0, 0, 0, 6, 4, 3, 0, 4, 0, 0, 0, 4, 3, 0, 4, 0, 0, 0, 0, 0, 6, 4, 2, 0, 0, 8, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 6, 4, 6, 4, 0, 4, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12

Offset: 1

Gary W. Adamson, Jan 15 2007

Left column = (1, 2, 2, 3, 2, 4, ...) = d(n), A000005; right border = (1, 1, 2, 2, 4, 2, 6, ...) = phi(n), A000010; row sums = (1, 3, 4, 7, 6, 12, ...) = sigma(n), A000203.

			First few rows of the triangle:
  1;
  2, 1;
  2, 0, 2;
  3, 2, 0, 2;
  2, 0, 0, 0, 4;
  4, 2, 4, 0, 0, 2;
  2, 0, 0, 0, 0, 0, 6;
  4, 3, 0, 4, 0, 0, 0, 4;
  3, 0, 4, 0, 0, 0, 0, 0, 6;
  4, 2, 0, 0, 8, 0, 0, 0, 0, 4;
  ...

Cf. A000005, A000010, A000203, A051731, A054522.

Edited by N. J. A. Sloane, Aug 11 2019

a(49) = 0 inserted and more terms from Georg Fischer, May 29 2023

A128184 A051731 * A097806.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 1, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Feb 17 2007

Row sums = A114003: (1, 3, 3, 5, 3, 7, 3, 7, 5, 7, ...).

			First few rows of the triangle:
  1;
  2, 1;
  1, 1, 1;
  2, 1, 1, 1;
  1, 0, 0, 1, 1;
  2, 2, 1, 0, 1, 1;
  ...

Cf. A051731, A097806, A114003.

A051731 * A097806, (inverse Moebius transform of A097806).

A130106 A051731 * diagonalized matrix of A063659.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 3, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 2, 0, 3, 0, 0, 0, 6, 1, 0, 3, 0, 0, 0, 0, 0, 8, 1, 2, 0, 0, 5, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 2, 3, 3, 0, 6, 0, 0, 0, 0, 0, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, May 07 2007

Right border = A063659, (1, 2, 3, 3, 5, 6, 7, 6, 8, 10, ...), the Moebius transform of A001615: (1, 3, 4, 6, 6, 12, 8, 12, 12, ...).
A130106 * (1, 2, 3, ...) = A034676: (1, 5, 10, 17, 26, 50, 50, ...).
A034676^(-1) * (1,2,3,...) = 1/1, 1/2, 2/3, 2/3, 4/5, 2/6, 6/7, 4/6, 6/8, 4/10, ...; where the numerators = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, 4, ...); and the denominators = A063659, the right border of the triangle: (1, 2, 3, 3, 5, 6, 7, 8, 10, ...).

			First few rows of the triangle:
  1;
  1, 2;
  1, 0, 3;
  1, 2, 0, 3;
  1, 0, 0, 0, 5;
  1, 2, 3, 0, 0, 6;
  1, 0, 0, 0, 0, 0, 7,
  1, 2, 0, 3, 0, 0, 0, 6;
  1, 0, 3, 0, 0, 0, 0, 0, 8;
  ...

Cf. A063659, A001615 (row sums), A051731, A000010.

m = 14;
A051731 = Table[If[Mod[n, k] == 0, 1, 0], {n, m}, {k, m}];
A063659 = Table[Sum[MoebiusMu[GCD[n, k]]^2, {k, n}], {n, m}] // DiagonalMatrix;
M = A051731.A063659;
Table[M[[n, k]], {n, m}, {k, n}] // Flatten (* Jean-François Alcover, Jan 18 2020 *)

Inverse Moebius transform of an infinite lower triangular matrix with A063659, (1, 2, 3, 3, 5, 6, 7, 6, 8, 10, ...) in the main diagonal and the rest zeros.

More terms from Jean-François Alcover, Jan 18 2020

A130160 A051731 * A130159 as a diagonalized matrix.

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 1, 0, 5, 1, 0, 0, 0, 7, 1, 1, 3, 0, 0, 6, 1, 0, 0, 0, 0, 0, 12, 1, 1, 0, 5, 0, 0, 0, 7, 1, 0, 3, 0, 0, 0, 0, 0, 12, 1, 1, 0, 0, 7, 0, 0, 0, 0, 10

Offset: 1

Gary W. Adamson, May 13 2007

Right border = A130159, the Moebius transform of the Odious numbers, A000069: (1, 2, 4, 7, 8, 11, ...) = (1, 1, 3, 5, 7, 6, 12, ...). Row sums = A000069.

			First few rows of the triangle:
  1;
  1,  1;
  1,  0,  3;
  1,  1,  0,  5;
  1,  0,  0,  0,  7;
  1,  1,  3,  0,  0,  6;
  1,  0,  0,  0,  0,  0, 12;
  ...

Cf. A130159, A000069, A051731.

Inverse Moebius transform (A051731) of an infinite lower triangular matrix with A130159 in the main diagonal and rest zeros.

A130210 Triangle read by rows: matrix product A051731 * A130209.

Original entry on oeis.org

1, 1, 2, 1, 0, 2, 1, 2, 0, 3, 1, 0, 0, 0, 2, 1, 2, 2, 0, 0, 4, 1, 0, 0, 0, 0, 0, 2, 1, 2, 0, 3, 0, 0, 0, 4, 1, 0, 2, 0, 0, 0, 0, 0, 3, 1, 2, 0, 0, 2, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 2, 3, 0, 4, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2

Offset: 1

Gary W. Adamson, May 17 2007

			First few rows of the triangle are:
1;
1, 2;
1, 0, 2;
1, 2, 0, 3;
1, 0, 0, 0, 2;
1, 2, 2, 0, 0, 4;
1, 0, 0, 0, 0, 0, 2;
...

Cf. A000005, A007425 (row sums).

A130210 := proc(n,k)
    add( A051731(n,j)*A130209(j,k),j=k..n) ;
end proc:
seq(seq(A130210(n,k),k=1..n),n=1..15) ;  # R. J. Mathar, Aug 06 2016

Inverse Moebius transform of A130209.

T(n,n) = A000005(n).

cp's OEIS Frontend

A160182 Triangle read by rows, 1 / ((-1)*A129184 * A051731 + I), I = Identity matrix.

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A174852 Triangle T(n,k) read by rows. T(1,1)=1, n>1 T(n,1)=A049240, k>1 T(n,k)=A051731.

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A177975 Square array T(n,k) read by antidiagonals up. Each column is the first column in the matrix inverse of a triangular matrix that is the k-th differences of A051731 in the column direction.

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A177978 Triangle T(n,k) read by rows: A051731(n,k) - A051731(n-1,k).

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A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

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A127471 Triangle formed from the matrix product A051731 * A054522 of infinite lower triangular matrices, read by rows.

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A128184 A051731 * A097806.

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A130106 A051731 * diagonalized matrix of A063659.

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A130160 A051731 * A130159 as a diagonalized matrix.

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A160182 Triangle read by rows, 1 / ((-1)A129184 A051731 + I), I = Identity matrix.