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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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A143325 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1) which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 6, 0, 1, 4, 15, 24, 15, 0, 1, 5, 24, 60, 80, 27, 0, 1, 6, 35, 120, 255, 232, 63, 0, 1, 7, 48, 210, 624, 1005, 728, 120, 0, 1, 8, 63, 336, 1295, 3096, 4095, 2160, 252, 0, 1, 9, 80, 504, 2400, 7735, 15624, 16320, 6552, 495, 0, 1, 10, 99
Offset: 1

Views

Author

Alois P. Heinz, Aug 07 2008

Keywords

Comments

Column k is Dirichlet convolution of mu(n) with k^(n-1). The coefficients of the polynomial of row n are given by the n-th row of triangle A054525; for example row 4 has polynomial -k+k^3.

Examples

			T(4,2)=6, because 6 words of length 4 over 2-letter alphabet {a,b} are primitive and earlier than others derived by cyclic shifts of the alphabet: aaab, aaba, aabb, abaa, abba, abbb; note that aaaa and abab are not primitive and words beginning with b can be derived by shifts of the alphabet from words in the list; secondly note that the words in the list need not be Lyndon words, for example aaba can be derived from aaab by a cyclic rotation of the positions.
Table begins:
  1,   1,    1,     1,     1,      1,      1,       1, ...
  0,   1,    2,     3,     4,      5,      6,       7, ...
  0,   3,    8,    15,    24,     35,     48,      63, ...
  0,   6,   24,    60,   120,    210,    336,     504, ...
  0,  15,   80,   255,   624,   1295,   2400,    4095, ...
  0,  27,  232,  1005,  3096,   7735,  16752,   32697, ...
  0,  63,  728,  4095, 15624,  46655, 117648,  262143, ...
  0, 120, 2160, 16320, 78000, 279720, 823200, 2096640, ...

Links

Crossrefs

Columns k=1-10 give: A063524, A000740, A034741, A295505, A295506, A320071, A320072, A320073, A320074, A320075.
Rows n=1-5, 7 give: A000012, A001477, A005563, A007531, A123865, A123866.
Main diagonal gives A075147.
Cf. A074650, A143324, A008683, A054525.

Programs

  • Maple
    with(numtheory):
f1:= proc(n) option remember;
       unapply(k^(n-1)-add(f1(d)(k), d=divisors(n)minus{n}), k)
     end;
T:= (n,k)-> f1(n)(k);
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);
  • Mathematica
    t[n_, k_] := Sum[k^(d-1)*MoebiusMu[n/d], {d, Divisors[n]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 21 2014, from first formula *)

Formula

T(n,k) = Sum_{d|n} k^(d-1) * mu(n/d).
T(n,k) = k^(n-1) - Sum_{d
T(n,k) = A074650(n,k) * n/k.
T(n,k) = A143324(n,k) / k.

A339684 a(n) = Sum_{d|n} 4^(d-1).

Original entry on oeis.org

1, 5, 17, 69, 257, 1045, 4097, 16453, 65553, 262405, 1048577, 4195413, 16777217, 67112965, 268435729, 1073758277, 4294967297, 17179935765, 68719476737, 274878169413, 1099511631889, 4398047559685, 17592186044417, 70368748389461, 281474976710913
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2020

Keywords

Links

Crossrefs

Cf. A000302, A295505.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), this sequence (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

Programs

  • Magma
    A339684:= func< n | (&+[4^(d-1): d in Divisors(n)]) >;
[A339684(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
  • Mathematica
    Table[Sum[4^(d - 1), {d, Divisors[n]}], {n, 1, 25}]
nmax = 25; CoefficientList[Series[Sum[x^k/(1 - 4 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, 4^(d-1)); \\ Michel Marcus, Dec 13 2020
  • SageMath
    def A339684(n): return sum(4^(k-1) for k in (1..n) if (k).divides(n))
[A339684(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

Formula

G.f.: Sum_{k>=1} x^k / (1 - 4*x^k).
G.f.: Sum_{k>=1} 4^(k-1) * x^k / (1 - x^k).
a(n) ~ 4^(n-1). - Vaclav Kotesovec, Jun 05 2021

A295506 a(n) = Sum_{d|n} mu(n/d)*5^(d-1).

Original entry on oeis.org

1, 4, 24, 120, 624, 3096, 15624, 78000, 390600, 1952496, 9765624, 48824880, 244140624, 1220687496, 6103514976, 30517500000, 152587890624, 762939059400, 3814697265624, 19073484374880, 95367431624976, 476837148437496, 2384185791015624, 11920928906172000
Offset: 1

Views

Author

Seiichi Manyama, Nov 23 2017

Keywords

Links

Crossrefs

Sum_{d|n} mu(n/d)*k^(d-1): A000740 (k=2), A034741 (k=3), A295505 (k=4), this sequence (k=5).
Column k=5 of A143325.
First differences of A320089.
Cf. A054720.

Programs

  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[Sum[MoebiusMu[k] * x^k / (1 - 5*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)
  • PARI
    {a(n) = sumdiv(n, d, moebius(n/d)*5^(d-1))}

Formula

a(n) = A054720(n)/5 for n > 0.
G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 5*x^k). - Ilya Gutkovskiy, Oct 25 2018

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
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