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.
%I A080383 #43 Sep 08 2022 08:45:09
%S A080383 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,
%T A080383 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,
%U A080383 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
%N 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).
%H A080383 Vaclav Kotesovec, <a href="/A080383/b080383.txt">Table of n, a(n) for n = 0..100000</a> (first 1000 terms from Vincenzo Librandi, terms 1001..9999 from David A. Corneth)
%e A080383 For n <= 500 only a few values of a(n) arise: {1,2,3,4,5,6,7,8,10,11,14}.
%e A080383 From _Jon E. Schoenfield_, Sep 15 2019: (Start)
%e A080383 a(n)=1 occurs only at n=0.
%e A080383 a(n)=2 occurs only at n=1.
%e A080383 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.)
%e A080383 a(n)=4 occurs only at n=3.
%e A080383 For n <= 100000, the only values of a(n) that occur are 1..16, 18, 19, 22, 23, and 26.
%e A080383 k | Indices n (up to 100000) at which a(n)=k
%e A080383 ---+-------------------------------------------------------
%e A080383 1 | 0
%e A080383 2 | 1
%e A080383 3 | 2, 4, 6, 8, 10, 14, 16, 18, 20, 22, 24, ...
%e A080383 4 | 3
%e A080383 5 | 40, 176, 208, 480, 736, 928, 1248, 1440, ... (A327430)
%e A080383 6 | 5, 7, 9, 11, 15, 17, 19, 21, 23, 27, 29, ... (A080384)
%e A080383 7 | 12, 30, 56, 84, 90, 132, 154, 182, 220, ... (A080385)
%e A080383 8 | 25, 37, 169, 199, 201, 241, 397, 433, ... (A080386)
%e A080383 9 | 1122, 1218, 5762, 11330, 12322, 15132, ... (A327431)
%e A080383 10 | 13, 31, 41, 57, 85, 91, 133, 155, 177, ... (A080387)
%e A080383 11 | 420, 920, 1892, 1978, 2444, 2914, 3198, ...
%e A080383 12 | 1103, 1703, 2863, 7773, 10603, 15133, ...
%e A080383 13 | 12324, 37444
%e A080383 14 | 421, 921, 1123, 1893, 1979, 1981, 2445, ...
%e A080383 15 | 4960, 6956, 13160, 16354, 18542, 24388, ...
%e A080383 16 | 11289, 16483, 36657, 62653, 89183
%e A080383 17 |
%e A080383 18 | 4961, 6957, 12325, 13161, 16355, 18543, ...
%e A080383 19 | 16356, 88510, 92004
%e A080383 20 |
%e A080383 21 |
%e A080383 22 | 16357, 88511, 90305, 92005
%e A080383 23 | 90306
%e A080383 24 |
%e A080383 25 |
%e A080383 26 | 90307
%e A080383 (End)
%t A080383 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 *)
%o A080383 (PARI) a(n) = my(b=binomial(n, n\2)); sum(i=0, n, (b % binomial(n, i)) == 0); \\ _Michel Marcus_, Jul 29 2017
%o A080383 (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
%o A080383 (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
%Y A080383 Cf. A001405, A000225, A057977, A022292, A020475, A067348, A042996.
%Y A080383 Cf. A327430, A080384, A080385, A080386, A327431, A080387.
%Y A080383 Cf. A080393.
%K A080383 nonn
%O A080383 0,2
%A A080383 _Labos Elemer_, Mar 12 2003
%E A080383 Edited by _Dean Hickerson_, Mar 14 2003
%E A080383 Offset corrected by _David A. Corneth_, Jul 29 2017