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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A320094 Number of primitive (=aperiodic) 10-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, 10, 109, 1099, 11098, 110989, 1110988, 11109988, 111109888, 1111099879, 11111099878, 111110998888, 1111110998887, 11111109998878, 111111109988779, 1111111099988779, 11111111099988778, 111111110999888878, 1111111110999888877, 11111111109999887887
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Links

Crossrefs

Column k=10 of A143327.
Partial sums of A320075.
Cf. A008683, A143325, A143326.

Programs

  • Maple
    b:= n-> add(`if`(d=n, 10^(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, 10^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 10^(d-1) * mu(j/d).
a(n) = A143327(n,10).
a(n) = Sum_{j=1..n} A143325(j,10).
a(n) = A143326(n,10) / 10.
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 10*x^k). - Ilya Gutkovskiy, Dec 11 2020

A320095 Number of primitive (=aperiodic) n-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, 2, 11, 79, 773, 9281, 137191, 2396150, 48426649, 1111099879, 28531150811, 810554312866, 25239591811405, 854769747700454, 31278135014945519, 1229782937960902111, 51702516367459973873, 2314494592652832016030, 109912203092221714132219, 5518821052631039996623577
Offset: 1

Views

Author

Alois P. Heinz, Oct 05 2018

Keywords

Links

Crossrefs

Main diagonal of A143327.
Cf. A143325, A143326.

Programs

  • Maple
    b:= (n, k)-> add(`if`(d=n, k^(n-1), -b(d, k)), d=numtheory[divisors](n)):
g:= proc(n, k) option remember; b(n, k)+`if`(n<2, 0, g(n-1, k)) end:
a:= n-> g(n$2):
seq(a(n), n=1..23);
  • Mathematica
    a[n_] := Sum[n^(d-1)*MoebiusMu[j/d], {j, 1, n}, {d, Divisors[j]}];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Oct 25 2022, after A143327 *)
  • PARI
    a(n) = sum(j=1, n, sumdiv(j, d, n^(d-1) * moebius(j/d))); \\ Michel Marcus, Feb 16 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} n^(d-1) * mu(j/d).
a(n) = A143327(n,n).
a(n) = Sum_{j=1..n} A143325(j,n).
a(n) = A143326(n,n) / n.
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - n*x^k). - Ilya Gutkovskiy, Feb 16 2020

A363916 Array read by descending antidiagonals. A(n, k) = Sum_{d=0..k} A363914(k, d) * n^d.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 6, 6, 4, 1, 0, 0, 12, 24, 12, 5, 1, 0, 0, 30, 72, 60, 20, 6, 1, 0, 0, 54, 240, 240, 120, 30, 7, 1, 0, 0, 126, 696, 1020, 600, 210, 42, 8, 1, 0, 0, 240, 2184, 4020, 3120, 1260, 336, 56, 9, 1
Offset: 0

Views

Author

Peter Luschny, Jul 04 2023

Keywords

Comments

Row n gives the number of n-ary sequences with primitive period k.
See A074650 and A143324 for combinatorial interpretations.

Examples

			Array A(n, k) starts:
[0] 1, 0,  0,   0,    0,     0,      0,       0,        0, ... A000007
[1] 1, 1,  0,   0,    0,     0,      0,       0,        0, ... A019590
[2] 1, 2,  2,   6,   12,    30,     54,     126,      240, ... A027375
[3] 1, 3,  6,  24,   72,   240,    696,    2184,     6480, ... A054718
[4] 1, 4, 12,  60,  240,  1020,   4020,   16380,    65280, ... A054719
[5] 1, 5, 20, 120,  600,  3120,  15480,   78120,   390000, ... A054720
[6] 1, 6, 30, 210, 1260,  7770,  46410,  279930,  1678320, ... A054721
[7] 1, 7, 42, 336, 2352, 16800, 117264,  823536,  5762400, ... A218124
[8] 1, 8, 56, 504, 4032, 32760, 261576, 2097144, 16773120, ... A218125
A000012|A002378| A047928   |   A218130     |      A218131
    A001477,A007531,    A061167,        A133499,   (diagonal A252764)
.
Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,  1;
[3] 0, 0,  2,   1;
[4] 0, 0,  2,   3,   1;
[5] 0, 0,  6,   6,   4,   1;
[6] 0, 0, 12,  24,  12,   5,  1;
[7] 0, 0, 30,  72,  60,  20,  6, 1;
[8] 0, 0, 54, 240, 240, 120, 30, 7, 1;

Links

Crossrefs

Variant: A143324.
Rows: A000007 (n=0), A019590 (n=1), A027375 (n=2), A054718 (n=3), A054719 (n=4), A054720, A054721, A218124, A218125.
Columns: A000012 (k=0), A001477 (k=1), A002378 (k=2), A007531(k=3), A047928, A061167, A218130, A133499, A218131.
Cf. A252764 (main diagonal), A074650, A363914.

Programs

  • Maple
    A363916 := (n, k) -> local d; add(A363914(k, d) * n^d, d = 0 ..k):
for n from 0 to 9 do seq(A363916(n, k), k = 0..8) od;
  • SageMath
    def A363916(n, k): return sum(A363914(k, d) * n^d for d in range(k + 1))
for n in range(9): print([A363916(n, k) for k in srange(9)])
def T(n, k): return A363916(k, n - k)

Formula

If k > 0 then k divides A(n, k), see the transposed array of A074650.
If k > 0 then n divides A(n, k), see the transposed array of A143325.
Previous Showing 21-23 of 23 results.

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