A080383 Number of j (0 <= j <= n) such that the central binomial coefficient C(n,floor(n/2)) = A001405(n) is divisible by C(n,j).

1, 2, 3, 4, 3, 6, 3, 6, 3, 6, 3, 6, 7, 10, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 8, 3, 6, 3, 6, 7, 10, 3, 6, 3, 6, 3, 8, 3, 6, 5, 10, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 7, 10, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 7, 10, 3, 6, 3, 6, 7, 10, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6

Offset: 0

Author

Labos Elemer, Mar 12 2003

Keywords

• nonn

Examples

			For n <= 500 only a few values of a(n) arise: {1,2,3,4,5,6,7,8,10,11,14}.
From _Jon E. Schoenfield_, Sep 15 2019: (Start)
a(n)=1 occurs only at n=0.
a(n)=2 occurs only at n=1.
a(n)=3 occurs for all even n > 0 such that C(n,j) divides C(n,n/2) only at j = 0, n/2, and n. (This is the case for about 4/9 of the first 100000 terms, and there appear to be nearly as many terms for which a(n)=6.)
a(n)=4 occurs only at n=3.
For n <= 100000, the only values of a(n) that occur are 1..16, 18, 19, 22, 23, and 26.
   k | Indices n (up to 100000) at which a(n)=k
  ---+-------------------------------------------------------
   1 | 0
   2 | 1
   3 | 2, 4, 6, 8, 10, 14, 16, 18, 20, 22, 24, ...
   4 | 3
   5 | 40, 176, 208, 480, 736, 928, 1248, 1440, ... (A327430)
   6 | 5, 7, 9, 11, 15, 17, 19, 21, 23, 27, 29, ... (A080384)
   7 | 12, 30, 56, 84, 90, 132, 154, 182, 220, ...  (A080385)
   8 | 25, 37, 169, 199, 201, 241, 397, 433, ...    (A080386)
   9 | 1122, 1218, 5762, 11330, 12322, 15132, ...   (A327431)
  10 | 13, 31, 41, 57, 85, 91, 133, 155, 177, ...   (A080387)
  11 | 420, 920, 1892, 1978, 2444, 2914, 3198, ...
  12 | 1103, 1703, 2863, 7773, 10603, 15133, ...
  13 | 12324, 37444
  14 | 421, 921, 1123, 1893, 1979, 1981, 2445, ...
  15 | 4960, 6956, 13160, 16354, 18542, 24388, ...
  16 | 11289, 16483, 36657, 62653, 89183
  17 |
  18 | 4961, 6957, 12325, 13161, 16355, 18543, ...
  19 | 16356, 88510, 92004
  20 |
  21 |
  22 | 16357, 88511, 90305, 92005
  23 | 90306
  24 |
  25 |
  26 | 90307
(End)
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..100000 (first 1000 terms from Vincenzo Librandi, terms 1001..9999 from David A. Corneth)

Crossrefs

Cf. A001405, A000225, A057977, A022292, A020475, A067348, A042996.

Cf. A327430, A080384, A080385, A080386, A327431, A080387.

Cf. A080393.

Programs

Magma

[#[j:j in [0..n]| Binomial(n,Floor(n/2)) mod Binomial(n,j) eq 0]:n in [0..100]]; // Marius A. Burtea, Sep 15 2019

Mathematica

Table[Count[Table[IntegerQ[Binomial[n, Floor[n/2]]/Binomial[n, j]], {j, 0, n}], True], {n, 0, 500}] (* adapted by Vincenzo Librandi, Jul 29 2017 *)

PARI

a(n) = my(b=binomial(n, n\2)); sum(i=0, n, (b % binomial(n, i)) == 0); \\ Michel Marcus, Jul 29 2017

PARI

a(n) = {if(n==0, return(1)); my(bb = binomial(n, n\2), b = n); res = 2 + !(n%2) + 2 * (n>2 && n%2 == 1); for(i = 2, (n-1)\2, res += 2*(bb%b==0); b *= (n + 1 - i) / i); res} \\ David A. Corneth, Jul 29 2017

Extensions

Edited by Dean Hickerson, Mar 14 2003

Offset corrected by David A. Corneth, Jul 29 2017