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.

Previous Showing 11-20 of 26 results. Next

A218131 Number of length 8 primitive (=aperiodic or period 8) n-ary words.

Original entry on oeis.org

0, 0, 240, 6480, 65280, 390000, 1678320, 5762400, 16773120, 43040160, 99990000, 214344240, 429960960, 815702160, 1475750640, 2562840000, 4294901760, 6975673920, 11019855600, 16983432720, 25599840000, 37822664880, 54875639280, 78310705440, 110074982400
Offset: 0

Views

Author

Alois P. Heinz, Oct 21 2012

Keywords

Links

Crossrefs

Row n=8 of A143324.
Cf. A090986, A013662, A267315.

Programs

  • Maple
    a:= n-> (n^4-1)*n^4:
seq(a(n), n=0..30);
  • Mathematica
    Table[n^8 - n^4, {n, 0, 30}] (* Wesley Ivan Hurt, Mar 30 2017 *)

Formula

G.f.: -240*x^2*(x+1)*(x^4+17*x^3+48*x^2+17*x+1)/(x-1)^9.
a(n) = n^8-n^4.
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=2} 1/a(n) = 15/8 - Pi^4/90 - Pi*coth(Pi)/4.
Sum_{n>=2} (-1)^n/a(n) = -7/8 + 7*Pi^4/720 - Pi*csch(Pi)/4 = -7/8 + A267315 - (1/4) * A090986. (End)

A252764 Number of length n primitive (=aperiodic or period n) n-ary words.

Original entry on oeis.org

1, 2, 24, 240, 3120, 46410, 823536, 16773120, 387419760, 9999899910, 285311670600, 8916097441680, 302875106592240, 11112006720144330, 437893890380096640, 18446744069414584320, 827240261886336764160, 39346408075098144278664, 1978419655660313589123960
Offset: 1

Views

Author

Alois P. Heinz, Dec 21 2014

Keywords

Examples

			a(3) = 24 because there are 24 primitive words of length 3 over 3-letter alphabet {a,b,c}: aab, aac, aba, abb, abc, aca, acb, acc, baa, bab, bac, bba, bbc, bca, bcb, bcc, caa, cab, cac, cba, cbb, cbc, cca, ccb.

Links

Crossrefs

Main diagonal of A143324.
Cf. A008683, A074650, A075147, A143324, A143325.

Programs

  • Maple
    with(numtheory):
a:= n-> add(n^d *mobius(n/d), d=divisors(n)):
seq(a(n), n=1..25);
  • Mathematica
    a[n_] := DivisorSum[n, n^# * MoebiusMu[n/#]& ];
Array[a, 25] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

Formula

a(n) = Sum_{d|n} n^d * mu(n/d), mu = A008683.
a(n) = A075147(n)*n.
a(n) = A074650(n,n) * n.
a(n) = A143325(n,n) * n.
a(n) = A143324(n,n).

A320071 Number of length n primitive (=aperiodic or period n) 6-ary words which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 5, 35, 210, 1295, 7735, 46655, 279720, 1679580, 10076395, 60466175, 362789070, 2176782335, 13060647355, 78364162765, 470184704640, 2821109907455, 16926657757380, 101559956668415, 609359729932590, 3656158440016285, 21936950579911675, 131621703842267135
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Comments

Dirichlet convolution of mu(n) with 6^(n-1).

Links

Crossrefs

Column k=6 of A143325.
First differences of A320090.
Cf. A008683, A074650, A143324.

Programs

  • Maple
    a:= n-> add(`if`(d=n, 6^(n-1), -a(d)), d=numtheory[divisors](n)):
seq(a(n), n=1..25);
  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[Sum[MoebiusMu[k] * x^k / (1 - 6*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)

Formula

a(n) = Sum_{d|n} 6^(d-1) * mu(n/d).
a(n) = 6^(n-1) - Sum_{d
a(n) = A143325(n,6).
a(n) = A074650(n,6) * n/6.
a(n) = A143324(n,6) / 6.
G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 6*x^k). - Ilya Gutkovskiy, Oct 25 2018

A320072 Number of length n primitive (=aperiodic or period n) 7-ary words which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 6, 48, 336, 2400, 16752, 117648, 823200, 5764752, 40351200, 282475248, 1977309600, 13841287200, 96888892752, 678223070400, 4747560686400, 33232930569600, 232630508205648, 1628413597910448, 11398895145019200, 79792266297494304, 558545863800808752
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Comments

Dirichlet convolution of mu(n) with 7^(n-1).

Links

Crossrefs

Column k=7 of A143325.
First differences of A320091.
Cf. A008683, A074650, A143324.

Programs

  • Maple
    a:= n-> add(`if`(d=n, 7^(n-1), -a(d)), d=numtheory[divisors](n)):
seq(a(n), n=1..25);

Formula

a(n) = Sum_{d|n} 7^(d-1) * mu(n/d).
a(n) = 7^(n-1) - Sum_{d
a(n) = A143325(n,7).
a(n) = A074650(n,7) * n/7.
a(n) = A143324(n,7) / 7.
G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 7*x^k). - Ilya Gutkovskiy, Oct 25 2018

A320073 Number of length n primitive (=aperiodic or period n) 8-ary words which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 7, 63, 504, 4095, 32697, 262143, 2096640, 16777152, 134213625, 1073741823, 8589901320, 68719476735, 549755551737, 4398046506945, 35184369991680, 281474976710655, 2251799796875328, 18014398509481983, 144115187941637640, 1152921504606584769
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Comments

Dirichlet convolution of mu(n) with 8^(n-1).

Links

Crossrefs

Column k=8 of A143325.
First differences of A320092.
Cf. A008683, A074650, A143324.

Programs

  • Maple
    a:= n-> add(`if`(d=n, 8^(n-1), -a(d)), d=numtheory[divisors](n)):
seq(a(n), n=1..25);

Formula

a(n) = Sum_{d|n} 8^(d-1) * mu(n/d).
a(n) = 8^(n-1) - Sum_{d
a(n) = A143325(n,8).
a(n) = A074650(n,8) * n/8.
a(n) = A143324(n,8) / 8.
G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 8*x^k). - Ilya Gutkovskiy, Oct 25 2018

A320074 Number of length n primitive (=aperiodic or period n) 9-ary words which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 8, 80, 720, 6560, 58960, 531440, 4782240, 43046640, 387413920, 3486784400, 31380999840, 282429536480, 2541865296880, 22876792448320, 205891127311680, 1853020188851840, 16677181656560880, 150094635296999120, 1350851717285570880, 12157665459056397280
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Comments

Dirichlet convolution of mu(n) with 9^(n-1).

Links

Crossrefs

Column k=9 of A143325.
First differences of A320093.
Cf. A008683, A074650, A143324.

Programs

  • Maple
    a:= n-> add(`if`(d=n, 9^(n-1), -a(d)), d=numtheory[divisors](n)):
seq(a(n), n=1..25);

Formula

a(n) = Sum_{d|n} 9^(d-1) * mu(n/d).
a(n) = 9^(n-1) - Sum_{d
a(n) = A143325(n,9).
a(n) = A074650(n,9) * n/9.
a(n) = A143324(n,9) / 9.
G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 9*x^k). - Ilya Gutkovskiy, Oct 25 2018

A218124 Number of 7-ary sequences with primitive period n.

Original entry on oeis.org

1, 7, 42, 336, 2352, 16800, 117264, 823536, 5762400, 40353264, 282458400, 1977326736, 13841167200, 96889010400, 678222249264, 4747561492800, 33232924804800, 232630513987200, 1628413557439536, 11398895185373136, 79792266015134400, 558545864082460128
Offset: 0

Views

Author

Alois P. Heinz, Oct 21 2012

Keywords

Links

Crossrefs

Column k=7 of A143324.

Programs

  • Maple
    with(numtheory):
a:= n-> `if`(n=0, 1, add(7^d *mobius(n/d), d=divisors(n))):
seq(a(n), n=0..30);

Formula

a(n) = Sum_{d|n} 7^d * mu(n/d) for n>0, a(0) = 1.
G.f.: 1 + 7 * Sum_{k>=1} mu(k) * x^k / (1 - 7*x^k). - Ilya Gutkovskiy, Apr 14 2021

A218125 Number of 8-ary sequences with primitive period n.

Original entry on oeis.org

1, 8, 56, 504, 4032, 32760, 261576, 2097144, 16773120, 134217216, 1073709000, 8589934584, 68719210560, 549755813880, 4398044413896, 35184372055560, 281474959933440, 2251799813685240, 18014398375002624, 144115188075855864, 1152921503533101120
Offset: 0

Views

Author

Alois P. Heinz, Oct 21 2012

Keywords

Links

Crossrefs

Column k=8 of A143324.

Programs

  • Maple
    with(numtheory):
a:= n-> `if`(n=0, 1, add(8^d*mobius(n/d), d=divisors(n))):
seq(a(n), n=0..30);

Formula

a(n) = Sum_{d|n} 8^d * mu(n/d) for n>0, a(0) = 1.
G.f.: 1 + 8 * Sum_{k>=1} mu(k) * x^k / (1 - 8*x^k). - Ilya Gutkovskiy, Apr 15 2021

A218130 Number of length 6 primitive (=aperiodic or period 6) n-ary words.

Original entry on oeis.org

0, 0, 54, 696, 4020, 15480, 46410, 117264, 261576, 530640, 998910, 1770120, 2984124, 4824456, 7526610, 11387040, 16772880, 24132384, 34006086, 47038680, 63991620, 85756440, 113368794, 148023216, 191088600, 244124400, 308897550, 387400104, 481867596, 594798120
Offset: 0

Views

Author

Alois P. Heinz, Oct 21 2012

Keywords

Links

Crossrefs

Row n=6 of A143324.

Programs

  • Maple
    a:= n-> (((n^3-1)*n-1)*n+1)*n:
seq(a(n), n=0..30);
  • Mathematica
    Table[n^6-n^3-n^2+n,{n,0,30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,54,696,4020,15480,46410},30] (* Harvey P. Dale, Oct 13 2017 *)

Formula

G.f.: -6*x^2*(9+53*x+47*x^2+11*x^3)/(x-1)^7.
a(n) = n^6-n^3-n^2+n.

A320075 Number of length n primitive (=aperiodic or period n) 10-ary words which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 9, 99, 990, 9999, 99891, 999999, 9999000, 99999900, 999989991, 9999999999, 99999899010, 999999999999, 9999998999991, 99999999989901, 999999990000000, 9999999999999999, 99999999899900100, 999999999999999999, 9999999998999999010, 99999999999998999901
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Comments

Dirichlet convolution of mu(n) with 10^(n-1).

Links

Crossrefs

Column k=10 of A143325.
First differences of A320094.
Cf. A008683, A074650, A143324.

Programs

  • Maple
    a:= n-> add(`if`(d=n, 10^(n-1), -a(d)), d=numtheory[divisors](n)):
seq(a(n), n=1..25);

Formula

a(n) = Sum_{d|n} 10^(d-1) * mu(n/d).
a(n) = 10^(n-1) - Sum_{d
a(n) = A143325(n,10).
a(n) = A074650(n,10) * n/10.
a(n) = A143324(n,10) / 10.
G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 10*x^k). - Ilya Gutkovskiy, Oct 25 2018
Previous Showing 11-20 of 26 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.