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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).
10
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
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