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.

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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

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

A320087 Number of primitive (=aperiodic) ternary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 3, 11, 35, 115, 347, 1075, 3235, 9787, 29387, 88435, 265315, 796755, 2390347, 7173227, 21519947, 64566667, 193700035, 581120523, 1743362283, 5230145947, 15690440099, 47071499707, 141214499227, 423644035627, 1270932113627, 3812797935395, 11438393826035
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Links

Crossrefs

Column k=3 of A143327.
Partial sums of A034741.
Cf. A008683, A143325, A143326.

Programs

  • Maple
    b:= n-> add(`if`(d=n, 3^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..30);
  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[1/(1-x) * Sum[MoebiusMu[k] * x^k / (1 - 3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)
  • PARI
    a(n) = sum(j=1, n, sumdiv(j, d, 3^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 3^(d-1) * mu(j/d).
a(n) = A143327(n,3).
a(n) = Sum_{j=1..n} A143325(j,3).
a(n) = A143326(n,3) / 3.
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 3*x^k). - Ilya Gutkovskiy, Dec 11 2020

A320088 Number of primitive (=aperiodic) 4-ary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 4, 19, 79, 334, 1339, 5434, 21754, 87274, 349159, 1397734, 5590954, 22368169, 89472934, 357908119, 1431633559, 5726600854, 22906403494, 91625880229, 366503524969, 1466015148634, 5864060611159, 23456246655574, 93824986622614, 375299963333014, 1501199853398419
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Links

Crossrefs

Column k=4 of A143327.
Partial sums of A295505.
Cf. A008683, A143325, A143326.

Programs

  • Maple
    b:= n-> add(`if`(d=n, 4^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..30);
  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[1/(1-x) * Sum[MoebiusMu[k] * x^k / (1 - 4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)
  • PARI
    a(n) = sum(j=1, n, sumdiv(j, d, 4^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 4^(d-1) * mu(j/d).
a(n) = A143327(n,4).
a(n) = Sum_{j=1..n} A143325(j,4).
a(n) = A143326(n,4) / 4.
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 4*x^k). - Ilya Gutkovskiy, Dec 11 2020

A320089 Number of primitive (=aperiodic) 5-ary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 5, 29, 149, 773, 3869, 19493, 97493, 488093, 2440589, 12206213, 61031093, 305171717, 1525859213, 7629374189, 38146874189, 190734764813, 953673824213, 4768371089837, 23841855464717, 119209287089693, 596046435527189, 2980232226542813, 14901161132714813
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Links

Crossrefs

Column k=5 of A143327.
Partial sums of A295506.
Cf. A008683, A143325, A143326.

Programs

  • Maple
    b:= n-> add(`if`(d=n, 5^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..30);
  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[1/(1-x) * Sum[MoebiusMu[k] * x^k / (1 - 5*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)
  • PARI
    a(n) = sum(j=1, n, sumdiv(j, d, 5^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 5^(d-1) * mu(j/d).
a(n) = A143327(n,5).
a(n) = Sum_{j=1..n} A143325(j,5).
a(n) = A143326(n,5) / 5.
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 5*x^k). - Ilya Gutkovskiy, Dec 11 2020

A320090 Number of primitive (=aperiodic) 6-ary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 6, 41, 251, 1546, 9281, 55936, 335656, 2015236, 12091631, 72557806, 435346876, 2612129211, 15672776566, 94036939331, 564221643971, 3385331551426, 20311989308806, 121871945977221, 731231675909811, 4387390115926096, 26324340695837771, 157946044538104906
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Links

Crossrefs

Column k=6 of A143327.
Partial sums of A320071.
Cf. A008683, A143325, A143326.

Programs

  • Maple
    b:= n-> add(`if`(d=n, 6^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..30);
  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[1/(1-x) * Sum[MoebiusMu[k] * x^k / (1 - 6*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)
  • PARI
    a(n) = sum(j=1, n, sumdiv(j, d, 6^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 6^(d-1) * mu(j/d).
a(n) = A143327(n,6).
a(n) = Sum_{j=1..n} A143325(j,6).
a(n) = A143326(n,6) / 6.
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 6*x^k). - Ilya Gutkovskiy, Dec 11 2020

A320091 Number of primitive (=aperiodic) 7-ary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 7, 55, 391, 2791, 19543, 137191, 960391, 6725143, 47076343, 329551591, 2306861191, 16148148391, 113037041143, 791260111543, 5538820797943, 38771751367543, 271402259573191, 1899815857483639, 13298711002502839, 93090977299997143, 651636841100805895
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Links

Crossrefs

Column k=7 of A143327.
Partial sums of A320072.
Cf. A008683, A143325, A143326.

Programs

  • Maple
    b:= n-> add(`if`(d=n, 7^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..30);
  • PARI
    a(n) = sum(j=1, n, sumdiv(j, d, 7^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 7^(d-1) * mu(j/d).
a(n) = A143327(n,7).
a(n) = Sum_{j=1..n} A143325(j,7).
a(n) = A143326(n,7) / 7.
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 7*x^k). - Ilya Gutkovskiy, Dec 11 2020

A320092 Number of primitive (=aperiodic) 8-ary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 8, 71, 575, 4670, 37367, 299510, 2396150, 19173302, 153386927, 1227128750, 9817030070, 78536506805, 628292058542, 5026338565487, 40210708557167, 321685685267822, 2573485482143150, 20587883991625133, 164703071933262773, 1317624576539847542
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Links

Crossrefs

Column k=8 of A143327.
Partial sums of A320073.
Cf. A008683, A143325, A143326.

Programs

  • Maple
    b:= n-> add(`if`(d=n, 8^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..30);
  • PARI
    a(n) = sum(j=1, n, sumdiv(j, d, 8^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 8^(d-1) * mu(j/d).
a(n) = A143327(n,8).
a(n) = Sum_{j=1..n} A143325(j,8).
a(n) = A143326(n,8) / 8.
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 8*x^k). - Ilya Gutkovskiy, Dec 11 2020

A320093 Number of primitive (=aperiodic) 9-ary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 9, 89, 809, 7369, 66329, 597769, 5380009, 48426649, 435840569, 3922624969, 35303624809, 317733161289, 2859598458169, 25736390906489, 231627518218169, 2084647707070009, 18761829363630889, 168856464660630009, 1519708181946200889, 13677373641002598169
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Links

Crossrefs

Column k=9 of A143327.
Partial sums of A320074.
Cf. A008683, A143325, A143326.

Programs

  • Maple
    b:= n-> add(`if`(d=n, 9^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..30);
  • PARI
    a(n) = sum(j=1, n, sumdiv(j, d, 9^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 9^(d-1) * mu(j/d).
a(n) = A143327(n,9).
a(n) = Sum_{j=1..n} A143325(j,9).
a(n) = A143326(n,9) / 9.
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 9*x^k). - Ilya Gutkovskiy, Dec 11 2020
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