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

0,2

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)

Example

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)

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 *)

Prog

(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
(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

Crossrefs

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

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

Cf. A080393.

Sequence in context: A288778 A290139 A317588 * A086369 A337532 A092089

Adjacent sequences: A080380 A080381 A080382 * A080384 A080385 A080386

Keyword

nonn

Author

Labos Elemer, Mar 12 2003

Extensions

Edited by Dean Hickerson, Mar 14 2003

Offset corrected by David A. Corneth, Jul 29 2017

Status

approved