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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A263673 a(n) = lcm{1,2,...,n} / binomial(n,floor(n/2)).

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 3, 12, 12, 20, 10, 60, 30, 210, 105, 56, 56, 504, 252, 2520, 1260, 660, 330, 3960, 1980, 5148, 2574, 4004, 2002, 30030, 15015, 240240, 240240, 123760, 61880, 31824, 15912, 302328, 151164, 77520, 38760, 813960, 406980, 8953560, 4476780, 2288132, 1144066, 27457584, 13728792, 49031400
Offset: 0

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Author

Max Alekseyev, Oct 23 2015

Keywords

Comments

From Robert Israel, Oct 23 2015: (Start)
If n = 2^k, a(n) = a(n-1).
If n = p^k where p is an odd prime and k >= 1, 2*n*a(n) = p*(n+1)*a(n-1).
If n is even and not a prime power, 2*a(n) = a(n-1).
If n is odd and not a prime power, 2*n*a(n) = (n+1)*a(n-1). (End)

Links

Crossrefs

Cf. A068550, A180000.

Programs

  • Maple
    a := n -> lcm(seq(k,k=1..n))/binomial(n,iquo(n,2)):
seq(a(n), n=0..49); # Peter Luschny, Oct 23 2015
  • Mathematica
    Join[{1}, Table[LCM @@ Range[n]/Binomial[n, Floor[n/2]], {n, 1, 50}]] (* or *) Table[Product[Cyclotomic[k, 1], {k, 2, n}]/Binomial[n, Floor[n/2]], {n, 0, 50}] (* G. C. Greubel, Apr 17 2017 *)
  • PARI
    A263673(n) = lcm(vector(n,i,i)) / binomial(n,n\2);

Formula

a(n) = A003418(n) / A001405(n).
a(n) = A048619(n-1) * A110654(n).
a(2*n) = A068550(n) = A099996(n) / A000984(n).
a(n) = A180000(n)*A152271(n). - Peter Luschny, Oct 23 2015
a(n) = (e/2)^(n + o(1)). - Charles R Greathouse IV, Oct 23 2015

A330332 a(n) = (number of times a(n-1) has already appeared) + (number of times a(n-2) has already appeared) + (number of times a(n-3) has already appeared), starting with a(n) = n for n<3.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 5, 6, 5, 7, 5, 9, 6, 7, 5, 9, 9, 11, 7, 7, 9, 12, 9, 11, 8, 8, 6, 7, 10, 9, 12, 9, 16, 10, 10, 7, 12, 12, 14, 9, 13, 10, 13, 8, 9, 14, 14, 15, 7, 11, 11, 15, 10, 11, 12, 15, 13, 11, 12, 15, 16, 12, 13, 13, 17, 11, 13, 14, 17, 12, 14, 15, 18, 11, 14, 15, 20, 13, 14, 15, 21, 15
Offset: 0

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Author

N. J. A. Sloane, Dec 14 2019

Keywords

Comments

Generalizes A316774, which looks at the frequencies of the two previous terms. Here we look at three previous terms.
If we look at just one previous term, we get 0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, ..., which is A133622 prefixed by 0, 1, or A152271 with its initial 1 changed to 0.

Links

Crossrefs

Cf. A316774, A133622, A152271.

Programs

  • Maple
    b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= `if`(n<3, n, b(a(n-1))+b(a(n-2))+b(a(n-3)));
      b(t):= b(t)+1; t
    end:
[seq(a(n), n=0..200)]; # Following Alois P. Heinz's program for A316774
  • Mathematica
    b[_] = 0;
a[n_] := a[n] = Module[{t}, t = If[n<3, n, b[a[n-1]] + b[a[n-2]] + b[a[n-3]]]; b[t]++; t];
a /@ Range[0, 200] (* Jean-François Alcover, Nov 09 2020, after Maple *)

A282738 First differences of A282737.

Original entry on oeis.org

1, 2, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 1, 16, 1, 17, 1, 18, 1, 19, 1, 20, 1, 21, 1, 22, 1, 23, 1, 24, 1, 25, 1, 26, 1, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 1, 39, 1, 40, 1, 41, 1
Offset: 0

Views

Author

N. J. A. Sloane, Mar 04 2017

Keywords

Links

Crossrefs

Cf. A282737.
A152271 and A133622 are very similar sequences.

Programs

  • PARI
    Vec((1 + 2*x - x^2 - x^5) / ((1 - x)^2*(1 + x)^2) + O(x^100)) \\ Colin Barker, Mar 04 2017

Formula

G.f.: (1 + 2*x - x^2 - x^5) / (1 - x^2)^2.
From Colin Barker, Mar 04 2017: (Start)
a(n) = 2*a(n-2) - a(n-4) for n>3.
a(n) = 1 for n>1 and even.
a(n) = (n+5) / 2 for n>1 and odd.
(End)

A362232 a(1) = 1; for n > 1, a(n) is number of terms in the first n-1 terms of the sequence that are not proper divisors of a(n-1).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 2, 4, 3, 6, 4, 6, 6, 7, 11, 12, 3, 14, 12, 5, 17, 18, 11, 20, 15, 19, 23, 24, 12, 15, 24, 14, 26, 28, 23, 32, 28, 26, 33, 32, 32, 33, 35, 38, 38, 39, 41, 44, 38, 43, 47, 48, 33, 46, 47, 52, 46, 50, 52, 49, 56, 48, 43, 60, 43, 62, 61, 64, 57, 63, 64, 60, 51, 67, 71, 72, 56, 64
Offset: 1

Views

Author

Scott R. Shannon, Apr 12 2023

Keywords

Examples

			a(8) = 2 as in the previous 8 - 1 = 7 terms there are two numbers that are not proper divisors of a(7) = 6, namely 4 and 6.

Links

Crossrefs

Cf. A027751, A032742, A032741, A152271.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1, (t-> add(
      `if`(irem(t, a(j))>0 or t=a(j), 1, 0), j=1..n-1))(a(n-1)))
    end:
seq(a(n), n=1..79);  # Alois P. Heinz, May 10 2023
  • Mathematica
    nn = 120; a[1] = 1; Do[Set[{c, m}, {0, a[n - 1]}]; Do[If[And[# < m, Divisible[m, #]] &[a[i]], c++], {i, n}]; a[n] = n - c - 1, {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, May 10 2023 *)
Previous Showing 11-14 of 14 results.

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

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