A327431 -id:A327431 - cp's OEIS frontend

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

Labos Elemer, Mar 12 2003

nonn

			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)

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

Cf. A001405, A000225, A057977, A022292, A020475, A067348, A042996.
Cf. A327430, A080384, A080385, A080386, A327431, A080387.
Cf. A080393.

[#[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

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 *)

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

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

Edited by Dean Hickerson, Mar 14 2003

Offset corrected by David A. Corneth, Jul 29 2017

A080385 Numbers k such that there are exactly 7 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 7.

Original entry on oeis.org

12, 30, 56, 84, 90, 132, 154, 182, 220, 252, 280, 306, 312, 340, 374, 380, 408, 418, 440, 456, 462, 476, 532, 552, 598, 616, 624, 630, 644, 650, 660, 690, 756, 828, 840, 858, 870, 880, 884, 900, 918, 936, 952, 966, 986, 992, 1020, 1054, 1102, 1116, 1140, 1160

Offset: 1

Labos Elemer, Mar 12 2003

nonn

			For n=12, the central binomial coefficient (C(12,6) = 924) is divisible by C(12,0), C(12,1), C(12,2), C(12,6), C(12,10), C(12,11), and C(12,12).

Vaclav Kotesovec, Table of n, a(n) for n = 1..4963

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

Cf. A001405, A057977.

More terms from Vaclav Kotesovec, Sep 06 2019

A080384 Numbers k such that there are exactly 6 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 6.

Original entry on oeis.org

5, 7, 9, 11, 15, 17, 19, 21, 23, 27, 29, 33, 35, 39, 43, 45, 47, 49, 51, 53, 55, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 87, 89, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 135, 137, 139, 141, 143, 145

Offset: 1

Labos Elemer, Mar 12 2003

nonn

			For n=9, the central binomial coefficient (C(9,4) = 126) is divisible by C(9,0), C(9,1), C(9,4), C(9,5), C(9,8), and C(9,9); certain primes are missing, certain composites are here.

Vaclav Kotesovec, Table of n, a(n) for n = 1..44084

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

Cf. A001405, A057977.

Position[Table[Count[Binomial[n,Floor[n/2]]/Binomial[n,Range[0,n]],?IntegerQ],{n,150}],6]//Flatten (* _Harvey P. Dale, Mar 05 2023 *)

isok(n) = my(b=binomial(n, n\2)); sum(i=0, n, (b % binomial(n, i)) == 0) == 6; \\ Michel Marcus, Jul 29 2017

A080386 Numbers k such that there are exactly 8 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 8.

Original entry on oeis.org

25, 37, 169, 199, 201, 241, 397, 433, 547, 685, 865, 1045, 1081, 1585, 1657, 1891, 1951, 1969, 2071, 2143, 2647, 2901, 3011, 3025, 3097, 3151, 3251, 3421, 3511, 3727, 4105, 4213, 4453, 4771, 4885, 5581, 5857, 6019, 6031, 6265, 6397, 6967, 7345, 7615, 7831, 8425, 8857, 8929

Offset: 1

Labos Elemer, Mar 12 2003

nonn

			For n=25, the central binomial coefficient (C(25,12) = 5200300) is divisible by C(25,0), C(25,1), C(25,3), C(25,12), C(25,13), C(25,22), C(25,24), and C(25,25).

Vaclav Kotesovec, Table of n, a(n) for n = 1..260

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

Cf. A001405, A057977.

More terms from Michel Marcus, Aug 23 2019

A080387 Numbers k such that there are exactly 10 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 10.

Original entry on oeis.org

13, 31, 41, 57, 85, 91, 133, 155, 177, 183, 209, 221, 253, 281, 307, 313, 341, 375, 381, 409, 419, 441, 457, 463, 477, 481, 533, 553, 599, 617, 625, 631, 645, 651, 661, 691, 737, 757, 829, 841, 859, 871, 881, 885, 901, 919, 929, 937, 953, 967, 987, 993

Offset: 1

Labos Elemer, Mar 12 2003

nonn

			For n=13, the central binomial coefficient (C(13,6) = 1716) is divisible by 10 binomial coefficients C(13,j); the 4 nondivisible cases are C(13,4), C(13,5), C(13,8), and C(13,9).

Vaclav Kotesovec, Table of n, a(n) for n = 1..5113

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

Cf. A001405, A057977.

A327430 Numbers k such that there are exactly 5 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 5.

Original entry on oeis.org

40, 176, 208, 480, 736, 928, 1248, 1440, 1632, 1824, 2128, 2400, 2464, 2720, 3008, 3360, 3520, 3776, 3904, 4144, 4240, 4320, 4704, 5280, 5664, 6432, 7040, 7200, 7360, 7488, 7992, 8064, 8544, 9504, 9792, 10336, 10400, 10944, 12160, 12992, 13158, 13392, 15744

Offset: 1

Vaclav Kotesovec, Sep 10 2019

nonn

			C(40,20) is divisible by 5 binomial coefficients: C(40,0), C(40,2), C(40,20), C(40,38) and C(40,40).

Vaclav Kotesovec, Table of n, a(n) for n = 1..181

Cf. A080383, A080384, A080385, A080386, A327431, A080387.

cp's OEIS Frontend

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).

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A080385 Numbers k such that there are exactly 7 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 7.

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A080384 Numbers k such that there are exactly 6 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 6.

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A080386 Numbers k such that there are exactly 8 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 8.

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A080387 Numbers k such that there are exactly 10 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 10.

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A327430 Numbers k such that there are exactly 5 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 5.

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