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 1-9 of 9 results.

A077602 Decimal expansion of lim_{n->inf} M(n,1)/2^n, where M(n,1) is the sum of the coefficients of the n-th Moebius polynomial (cf. A074587).

Original entry on oeis.org

1, 5, 3, 0, 1, 9, 1, 4, 1, 4, 0, 1, 6, 5, 4, 9, 1, 8, 7, 1, 5, 4, 3, 6, 2, 3, 6, 1, 4, 9, 2, 6, 3, 3, 0, 2, 0, 2, 5, 9, 5, 1, 2, 3, 7, 4, 1, 1, 1, 5, 7, 1, 0, 0, 7, 0, 7, 0, 6, 0, 1, 1, 1, 3, 9, 3, 1, 7, 5, 3, 5, 5, 9, 5, 7, 1, 3, 7, 3, 1, 1, 3, 9, 8, 8, 1, 2
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

Keywords

Comments

Conjecture: M(n,1) ~ A077596(n) * sqrt(Pi*n/2), where A077596(n) is the largest coefficient of the n-th Moebius polynomial, M(n,x).

Examples

			1.530191414016549187154362361492633020259512374111571007070601113931753...

Links

Crossrefs

Cf. A074586, A074587, A077596, A077598, A077599, A077600, A077601.

Programs

  • Mathematica
    Clear[Moebius,f]; Moebius[n_, x_] := Moebius[n, x] = 1 + x*Sum[Moebius[k, x]*Floor[n/k], {k, 1, n-1}]; f[n_] := f[n] = RealDigits[Moebius[n, 1]/2^n, 10, 70] // First; f[n=100]; While[f[n] != f[n-100], n = n+100]; f[n] (* Jean-François Alcover, Feb 13 2013 *)

A074586 Triangle of Moebius polynomial coefficients, read by rows, the n-th row forming the polynomial M(n,x) such that M(n,-1) = mu(n), the Moebius function of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 7, 8, 2, 1, 9, 15, 10, 2, 1, 13, 30, 27, 12, 2, 1, 15, 43, 57, 39, 14, 2, 1, 19, 67, 108, 98, 53, 16, 2, 1, 22, 90, 177, 206, 151, 69, 18, 2, 1, 26, 123, 282, 393, 359, 220, 87, 20, 2, 1, 28, 149, 405, 675, 752, 579, 307, 107, 22, 2, 1, 34, 203, 594, 1109
Offset: 1

Views

Author

Paul D. Hanna, Aug 25 2002

Keywords

Examples

			The first few Moebius polynomials are as follows:
M(1,x) = 1;
M(2,x) = 1 + 2*x;
M(3,x) = 1 + 4*x + 2*x^2;
M(4,x) = 1 + 7*x + 8*x^2 + 2*x^3;
M(5,x) = 1 + 9*x + 15*x^2 + 10*x^3 + 2*x^4;
M(6,x) = 1 + 13*x + 30*x^2 + 27*x^3 + 12*x^4 + 2*x^5;
M(7,x) = 1 + 15*x + 43*x^2 + 57*x^3 + 39*x^4 + 14*x^5 + 2*x^6; ...
ILLUSTRATION OF GENERATING METHOD:
M(1,x) = 1;
M(2,x) = 1 + 2*x*M(1,x) = 1 + 2*x;
M(3,x) = 1 + 3*x*M(1,x) + [3/2]*x*M(2,x) = 1 + 3*x + x*(1+2*x) = 1 + 4*x + 2*x^2;
M(4,x) = 1 + 4*x*M(1,x) + [4/2]*x*M(2,x) + [4/3]*x*M(3,x) = 1 + 4*x + 2*x*(1 + 2*x) + 1*x*(1 + 4*x + 2*x^2) = 1 + 7*x + 8*x^2 + 2*x^3;
M(5,x) = 1 + 5*x*M(1,x) + [5/2]*x*M(2,x) + [5/3]*x*M(3,x) + [5/4]*x*M(4,x) = 1 + 5*x + 2*x*(1 + 2*x) + 1*x*(1 + 4*x + 2*x^2) + 1*x*(1 + 7*x + 8*x^2 + 2*x^3) = 1 + 9*x + 15*x^2 + 10*x^3 + 2*x^4; ...
This triangle of coefficients begins:
  1
  1  2
  1  4   2
  1  7   8   2
  1  9  15  10    2
  1 13  30  27   12    2
  1 15  43  57   39   14    2
  1 19  67 108   98   53   16   2
  1 22  90 177  206  151   69  18   2
  1 26 123 282  393  359  220  87  20   2
  1 28 149 405  675  752  579 307 107  22  2
  1 34 203 594 1109 1439 1333 886 414 129 24 2 ...

Links

Crossrefs

Cf. A074587.

Programs

  • Mathematica
    t[n_, 1] = 1; t[n_, k_] := t[n, k] = Sum[ Floor[n/m]*t[m, k-1], {m, 1, n-1}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 03 2012, after PARI *)
  • PARI
    {T(n,k)=if(k==1,1,sum(m=1,n-1,floor(n/m)*T(m,k-1)))}
for(n=1,12, for(k=1,n, print1(T(n,k),", "));print(""))

Formula

The n-th row consists of the coefficients of M(n, x) as a polynomial in x, where M(n, x) = 1 + [n/1]*x*M(1, x) + [n/2]*x*M(2, x) + [n/3]*x*M(3, x) +... + [n/(n-1)]*x*M(n-1, x) for n>1, with M(1, x) = 1, where [x] = floor(x).
T(n, k) = Sum_{m=1..n-1} [n/m]*T(m, k-1) for n>=k>1, with T(n, 1)=1 for n>=1.

A077596 Central coefficients of Moebius polynomials (A074586): coefficient of x^(n/2-1/2) if n is odd; coefficient of x^(n/2-1) if n is even and >4. The n-th Moebius polynomial, M(n,x), satisfies M(n,-1)=mu(n) the Moebius function of n.

Original entry on oeis.org

1, 2, 4, 8, 15, 30, 57, 108, 206, 393, 752, 1439, 2772, 5334, 10327, 19967, 38808, 75319, 146844, 285862, 558723, 1090370, 2135551, 4176224, 8193490, 16050930, 31537017, 61872863, 121721157, 239115024, 470918888, 926141652, 1825708221
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

Keywords

Comments

These terms seem to be asymptotic to c*2^n/sqrt(n) with c=1.2208..
c = 1.220916104316909855089768170983761594215082355524... . - Vaclav Kotesovec, Feb 11 2015

Examples

			These are the largest coefficients of the Moebius polynomials, which begin:
M(1,x) = 1;
M(2,x) = 1 + 2x;
M(3,x) = 1 + 4x + 2x^2;
M(4,x) = 1 + 7x + 8x^2 + 2x^3;
M(5,x) = 1 + 9x +15x^2 +10x^3 + 2x^4;
M(6,x) = 1 +13x +30x^2 +27x^3 +12x^4 + 2x^5;
M(7,x) = 1 +15x +43x^2 +57x^3 +39x^4 +14x^5 + 2x^6;
M(8,x) = 1 +19x +67x^2+108x^3 +98x^4 +53x^5 +16x^6 + 2x^7; ...

Crossrefs

Cf. A074586, A074587, A077597, A077598, A077599, A077600, A077601.

Programs

  • Mathematica
    m[n_, 1] = 1; m[n_, k_] := m[n, k] = Sum[Floor[n/j]*m[j, k - 1], {j, 1, n - 1}];
a[n_ /; n <= 4] := 2^(n - 1); a[n_?OddQ] := m[n, (n + 1)/2]; a[n_?EvenQ] := m[n, n/2]; Table[a[n], {n, 1, 33}] (* Jean-François Alcover, Jun 18 2013 *)

A077597 Coefficient of x in the n-th Moebius polynomial (A074586), M(n,x), which satisfies M(n,-1)=mu(n) the Moebius function of n.

Original entry on oeis.org

0, 2, 4, 7, 9, 13, 15, 19, 22, 26, 28, 34, 36, 40, 44, 49, 51, 57, 59, 65, 69, 73, 75, 83, 86, 90, 94, 100, 102, 110, 112, 118, 122, 126, 130, 139, 141, 145, 149, 157, 159, 167, 169, 175, 181, 185, 187, 197, 200, 206, 210, 216, 218, 226, 230, 238, 242, 246, 248, 260
Offset: 0

Views

Author

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

Keywords

Comments

This is also the number of ways to misidentify a solar mode of degree l with modes of lower degree. See paper with Lou Lanzerotti (in preparation). - David J. Thomson, Oct 28 2010

Examples

			These are the coefficients of x in the Moebius polynomials, which begin: M(1,x) = 1; M(2,x) = 1 + 2x; M(3,x) = 1 + 4x + 2x^2; M(4,x) = 1 + 7x + 8x^2 + 2x^3; M(5,x) = 1 + 9x + 15x^2 + 10x^3 + 2x^4; M(6,x) = 1 + 13x + 30x^2 + 27x^3 + 12x^4 + 2x^5; M(7,x) = 1 + 15x + 43x^2 + 57x^3 + 39x^4 + 14x^5 + 2x^6; M(8,x) = 1 + 19x + 67x^2 + 108x^3 + 98x^4 + 53x^5 + 16x^6 + 2x^7.

Links

Crossrefs

Cf. A074586, A074587, A077596, A077598, A077599, A077600, A077601, A085683.
Equals A006218(n+1) - 1.

Programs

  • Mathematica
    a[n_] := Sum[ Floor[(n+1)/k], {k, 1, n+1}] - 1; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Jun 18 2013 *)

Formula

a(n) = Sum_{k = 1..n} floor((n+1)/k). - N. J. A. Sloane, Oct 28 2008
Since a(n) = A006218(n+1) - 1, asymptotics and bounds may be obtained from that entry.

A077598 Coefficient of x^2 in the n-th Moebius polynomial (A074586), M(n,x), which satisfies M(n,-1)=mu(n) the Moebius function of n.

Original entry on oeis.org

0, 0, 2, 8, 15, 30, 43, 67, 90, 123, 149, 203, 237, 290, 343, 415, 464, 556, 613, 716, 800, 899, 972, 1126, 1218, 1342, 1458, 1616, 1716, 1916, 2026, 2215, 2365, 2540, 2690, 2959, 3098, 3300, 3485, 3762, 3919, 4221, 4388, 4667, 4921, 5179, 5364, 5762
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

Keywords

Comments

These terms seem to be asymptotic to c*n^2*log(n) with c=0.69...

Examples

			These are the coefficients of x^2 in the Moebius polynomials, which begin: M(1,x)=1; M(2,x)=1 + 2x; M(3,x)=1 + 4x + 2x^2; M(4,x)=1 + 7x + 8x^2 + 2x^3; M(5,x)=1 + 9x +15x^2 +10x^3 + 2x^4; M(6,x)=1 +13x +30x^2 +27x^3 +12x^4 + 2x^5; M(7,x)=1 +15x +43x^2 +57x^3 +39x^4 +14x^5 + 2x^6; M(8,x)=1 +19x +67x^2+108x^3 +98x^4 +53x^5 +16x^6 + 2x^7.

Crossrefs

Cf. A074586, A074587, A077596, A077597, A077599, A077600, A077601.

Programs

  • Mathematica
    m[n_, 1] = 1; m[n_, k_] := m[n, k] = Sum[Floor[n/j]*m[j, k - 1], {j, 1, n - 1}]; a[n_] := m[n, 3]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Jun 18 2013 *)

A077599 Sequence of n such that -1 is a "double" root for M(n,x) (i.e., M(n,x)=(x+1)^2*Q(n,x)).

Original entry on oeis.org

8, 9, 24, 45, 100, 117, 120, 125, 135, 171, 175, 180, 184, 224, 243, 248, 256, 261, 270, 304, 312, 324, 342, 343, 344, 360, 369, 405, 459, 468, 472, 475, 477, 486, 507, 513, 520, 531, 536, 578, 584, 603, 608, 625, 639, 640, 657, 664, 675, 704, 711, 720, 728
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

Keywords

Comments

The n-th Moebius polynomial M(n,x) satisfies M(n,-1)=mu(n), the Moebius function of n; thus -1 is a simple root of M(n,x) if n is not squarefree. Hence these values could be called "double nonsquarefree numbers".
The n-th polynomial is divisible by (x+1)^3 for n=175, 343, 513, 800, 875. - T. D. Noe, Jan 09 2008

Crossrefs

Cf. A074586, A074587, A077596, A077597, A077598, A077600, A077601.

Programs

  • Mathematica
    a[n_,1]=1; a[n_,k_]:=a[n, k]=Sum[Floor[n/m] a[m,k-1], {m,n-1}]; t={}; Do[p=Table[a[n,k], {k,n}].(x^Range[0,n-1]); If[PolynomialMod[p,(x+1)^2]==0, AppendTo[t,n]], {n,100}]; t (* T. D. Noe, Jan 09 2008 *)

Extensions

More terms from T. D. Noe, Jan 09 2008

A077600 Number of real roots for the n-th Moebius polynomial, M(n,x), which satisfies M(n,-1)=mu(n) the Moebius function of n.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 2, 5, 4, 3, 2, 5, 4, 3, 4, 5, 4, 5, 2, 5, 4, 3, 2, 7, 6, 5, 6, 5, 4, 5, 4, 7, 4, 5, 4, 7, 6, 5, 6, 7, 4, 5, 4, 5, 6, 5, 4, 7, 6, 7, 6, 5, 4, 7, 4, 5, 4, 5, 4, 7, 6, 5, 8, 7, 6, 5, 6, 9, 6, 5, 4, 9, 8, 5, 8, 9, 4, 5, 4, 7, 8, 7, 4, 7, 6, 7, 6, 7, 4, 9, 4, 7, 8, 5, 6, 7, 6, 9, 8
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

Keywords

Crossrefs

Cf. A074586, A074587, A077596, A077597, A077598, A077599, A077601.

Programs

  • Mathematica
    m[1, x_] = 1; m[n_, x_] :=  m[n, x] = 1 + Sum[x*m[k, x]*Floor[n/k], {k, 1, n-1}] // Expand; a[n_] := CountRoots[m[n, x], x]; Table[a[n], {n, 1, 99}] (* Jean-François Alcover, Sep 13 2012 *)

Extensions

Typo (?) a(45)=4 replaced with 6 by Jean-François Alcover, Sep 13 2012

A077601 Decimal expansion of the limit of the maximum real root of M(n,-x) as n -> infinity, where M(n,x) is the n-th Moebius polynomial and satisfies M(n,-1) = mu(n) the Moebius function of n.

Original entry on oeis.org

2, 9, 8, 7, 4, 5, 8, 1, 3, 6, 6, 8, 7, 2, 2, 0, 0, 7
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Nov 10 2002

Keywords

Examples

			Let R(n) = max zero of M(n,-x), then R(5) = 2.961148315..., R(10)= 2.980078049..., R(15)= 2.986397371..., R(oo)= 2.98745813668722007...

Crossrefs

Cf. A074586, A074587, A077596, A077598, A077599, A077600, A077602.

Programs

  • Mathematica
    digits = 18; t[n_, 1] = 1; t[n_, k_] := t[n, k] = Sum[ Floor[n/m]*t[m, k-1], {m, 1, n-1}]; m[n_, x_] := Sum[ t[n, k+1]*x^k, {k, 0, n}]; Clear[f]; f[n_] := f[n] = Table[ Root[ m[n, -x], k, x], {k, 1, n-1}] // N[#, digits+5]& // Re // Max; Catch[ For[ n = 3, True, n++, If[ RealDigits[ f[n]][[1]][[1 ;; digits+2]] == RealDigits[ f[n-1]][[1]][[1 ;; digits+2]], Throw[f[n]]]]] // RealDigits[#, 10, digits+1]& // First (* Jean-François Alcover, Apr 12 2013 *)

A169659 Triangle T(n,k) read by rows. Matrix inverse of A174820.

Original entry on oeis.org

1, 3, 1, 7, 2, 1, 18, 5, 2, 1, 37, 10, 4, 2, 1, 85, 23, 9, 4, 2, 1, 171, 46, 18, 8, 4, 2, 1, 364, 98, 38, 17, 8, 4, 2, 1, 736, 198, 77, 34, 16, 8, 4, 2, 1, 1513, 407, 158, 70, 33, 16, 8, 4, 2, 1, 3027, 814, 316, 140, 66, 32, 16, 8, 4, 2, 1, 6168, 1659, 644, 285, 134, 65, 32, 16, 8, 4
Offset: 1

Views

Author

Mats Granvik, Paul D. Hanna, Apr 05 2010

Keywords

Examples

			Table begins:
...1
...3....1
...7....2....1
..18....5....2....1
..37...10....4....2....1
..85...23....9....4....2....1
.171...46...18....8....4....2....1
.364...98...38...17....8....4....2....1
.736..198...77...34...16....8....4....2....1
1513..407..158...70...33...16....8....4....2....1
3027..814..316..140...66...32...16....8....4....2....1
6168.1659..644..285..134...65...32...16....8....4....2....1

Crossrefs

Cf. A174820, A074587.
Showing 1-9 of 9 results.

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