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.

Showing 1-5 of 5 results.

A067348 Even numbers n such that binomial(n, [n/2]) is divisible by n.

Original entry on oeis.org

2, 12, 30, 56, 84, 90, 132, 154, 182, 220, 252, 280, 306, 312, 340, 374, 380, 408, 418, 420, 440, 456, 462, 476, 532, 552, 598, 616, 624, 630, 644, 650, 660, 690, 756, 828, 840, 858, 870, 880, 884, 900, 918, 920, 936, 952, 966, 986, 992, 1020, 1054, 1102
Offset: 1

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Author

Dean Hickerson, Jan 16 2002

Keywords

Comments

This sequence has a surprisingly large overlap with A080385(n); a few values, 2, 420, 920 are exceptional. This means that usually A080383(A067348(n))=7. - Labos Elemer, Mar 17 2003
Conjecture: sequence contains most of 2*A000384(k). Exceptions are k = 8, 18, 20, 23, 35, ... - Ralf Stephan, Mar 15 2004

Links

Crossrefs

Subsequence of A042996.
Cf. A000984, A067315, A080385.

Programs

  • Mathematica
    Select[Range[2, 1200, 2], Mod[Binomial[ #, #/2], # ]==0&]
  • PARI
    val(n, p) = my(r=0); while(n, r+=n\=p);r
is(n) = {if(valuation(n, 2) == 0, return(0)); my(f = factor(n)); for(i=1, #f~, if(val(n, f[i, 1]) - 2 * val(n/2, f[i, 1]) - f[i, 2] < 0, return(0))); return(1)} \\ David A. Corneth, Jul 29 2017

Formula

a(n) = 2*A002503(n-2) + 2.
Appears to be 2*A058008(n). - Benoit Cloitre, Mar 21 2003

Extensions

Name clarified by Peter Luschny, Aug 04 2017

A099905 a(n) = binomial(2n-1, n-1) mod n.

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 3, 1, 8, 1, 2, 1, 10, 0, 3, 1, 12, 1, 10, 3, 14, 1, 6, 1, 16, 10, 0, 1, 2, 1, 3, 21, 20, 21, 26, 1, 22, 10, 10, 1, 0, 1, 24, 0, 26, 1, 30, 1, 28, 27, 48, 1, 30, 16, 44, 48, 32, 1, 48, 1, 34, 6, 35, 35, 0, 1, 18, 33, 20, 1, 18, 1, 40, 60, 16, 0, 72, 1, 10, 10, 44, 1, 56, 75
Offset: 1

Views

Author

Henry Bottomley, Oct 29 2004

Keywords

Comments

For p prime, a(p)=1. For n in A058008, a(n)=0.
For n the square of a prime p>=3 or the cube of a prime p>=5, a(n)=1. - Franz Vrabec, Mar 26 2008
For n in A228562, a(n)=1. - Felix Fröhlich, Oct 17 2015

Examples

			a(11) = 352716 mod 11 = 1.

Links

Crossrefs

Cf. A058008, A088218, A099906, A099907, A099908, A228562.

Programs

A075055 Smallest integer of the form product (n+1)(n+2)...(n+k)/n!.

Original entry on oeis.org

1, 1, 6, 20, 70, 252, 7, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 178296, 601080390, 2333606220, 9075135300, 35345263800, 137846528820, 538257874440, 2104098963720, 8233430727600, 32247603683100, 126410606437752, 495918532948104
Offset: 0

Views

Author

Amarnath Murthy, Sep 07 2002

Keywords

Comments

Differs from A000984 at positions in A058008.

Examples

			a(6) = 7 = 7*8*9*10/6! = 5040/720.

Links

Crossrefs

Cf. A075054.

Programs

  • Maple
    a:= proc(n) option remember; local k, t; t:= 1/n!;
      for k while denom(t)>1 do t:= t*(n+k) od; t
    end:
seq(a(n), n=0..28);  # Alois P. Heinz, Feb 05 2025
  • Mathematica
    a[n_] := Catch[ For[ k = n-2, True, k++, If[ IntegerQ[an = (n+k)!/n!^2], Throw[an]]]]; a[1]=1; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Jun 28 2012 *)

Extensions

More terms from Sascha Kurz, Feb 02 2003
a(0)=1 prepended by Alois P. Heinz, Feb 05 2025

A075054 Smallest k such that (n+1)(n+2)...(n+k) is divisible by n!.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 8, 9, 10, 11, 12, 13, 14, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 40, 43, 44, 43, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 64, 67, 68, 69, 70, 71, 72
Offset: 1

Views

Author

Amarnath Murthy, Sep 07 2002

Keywords

Comments

a(n) <= n. a(n) < n rarely, e.g. for n = 6, 15 etc. a(p) = p, p is a prime.

Examples

			a(6) = 4 as 7*8*9*10 is divisible by 6!= 720.

Links

Crossrefs

Cf. A075055, A058008.

Programs

  • Mathematica
    dnf[n_]:=Module[{nf=n!,k=1},While[!Divisible[Times@@Range[ n+1,n+k],nf],k++];k]; Array[dnf,80] (* Harvey P. Dale, Jun 19 2012 *)

Extensions

More terms from Sascha Kurz, Feb 02 2003
Edited by Charles R Greathouse IV, Aug 02 2010

A215619 a(n) is the number of consecutive terms of A100071, beginning with index n, which are divisible by n.

Original entry on oeis.org

4, 1, 6, 1, 8, 1, 4, 1, 12, 5, 14, 1, 4, 1, 18, 1, 20, 1, 4, 1, 24, 1, 6, 1, 4, 1, 30, 21, 32, 1, 12, 1, 8, 1, 38, 1, 14, 1, 42, 1, 44, 1, 6, 1, 48, 1, 8, 1, 4, 1, 54, 1, 6, 9, 4, 1, 60, 1, 62, 1, 4, 1, 6, 1, 68, 1, 4, 1, 72, 1, 74, 1, 4, 1, 12, 1, 80, 1, 4, 1
Offset: 3

Views

Author

Vladimir Shevelev, Aug 17 2012

Keywords

Comments

a(n) = n+1 iff n is prime.
a(n) = 1 iff n in { A067315 }.
1 <= a(n) <= n+1.
{ n : a(2n)>1 } = { A058008 } \ { 1 }.

Links

Crossrefs

Cf. A000040, A067315, A100071.

Programs

  • Maple
    b:= proc(n) b(n):= n * binomial(n-1, floor((n-1)/2)) end:
a:= proc(n) local k;
      for k from 0 while irem(b(n+k), n)=0 do od; k
    end:
seq (a(n), n=3..100);  # Alois P. Heinz, Aug 17 2012
  • Mathematica
    b[n_] := n Binomial[n-1, Floor[(n-1)/2]];
a[n_] := Module[{k = 0}, While[Mod[b[n+k], n] == 0, k++]; k];
a /@ Range[3, 100] (* Jean-François Alcover, Nov 22 2020, after Alois P. Heinz *)
Showing 1-5 of 5 results.

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