A067348 -id:A067348 - 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

A002503 Numbers k such that binomial(2*k,k) is divisible by (k+1)^2.

Original entry on oeis.org

5, 14, 27, 41, 44, 65, 76, 90, 109, 125, 139, 152, 155, 169, 186, 189, 203, 208, 209, 219, 227, 230, 237, 265, 275, 298, 307, 311, 314, 321, 324, 329, 344, 377, 413, 419, 428, 434, 439, 441, 449, 458, 459, 467, 475

Offset: 1

N. J. A. Sloane, Mira Bernstein

From Amiram Eldar, Mar 28 2021: (Start)
Balakram (1929) proved that:
1) This sequence is infinite.
2) If m is an even perfect number (A000396) then m-1 is a term.
3) If m = p*q - 1, where p and q are primes, and (3/2)*p < q < 2*p, then m is a term.
4) m is a term if and only if Sum_{k>=1} floor(2*m/p^k) >= 2 * Sum_{k>=1} floor((m+1)/p^k), for all primes p. (End)

Hoon Balakram, On the values of n which make (2n)!/(n+1)!(n+1)! an integer, J. Indian Math. Soc., Vol. 18 (1929), pp. 97-100.
Thomas Koshy, Catalan numbers with applications, Oxford University Press, 2008, pp. 69-70.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp., Vol. 29, No. 129 (1975), pp. 83-92.
Carl Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, Vol. 112, No. 7 (2015), pp. 636-644; alternative link.

Positions of zeros in A065350.
Cf. A000108, A065344-A065349.
Equals A067348(n+2)/2 - 1.
Cf. A000396, A135627.

import Data.List (elemIndices)
a002503 n = a002503_list !! (n-1)
a002503_list = map (+ 1) $ elemIndices 0 a065350_list
-- Reinhard Zumkeller, Sep 16 2014

Select[Range[500],Divisible[Binomial[2#,#],(#+1)^2]&] (* Harvey P. Dale, May 21 2012 *)

isok(n) = binomial(2*n, n) % (n+1)^2 == 0; \\ Michel Marcus, Jan 11 2016

A065350(a(n)) = 0. - Reinhard Zumkeller, Sep 16 2014

Balakram reference corrected by T. D. Noe, Jan 16 2007

A058008 Numbers k such that (2*k - 1)!/(k!)^2 is an integer.

Original entry on oeis.org

1, 6, 15, 28, 42, 45, 66, 77, 91, 110, 126, 140, 153, 156, 170, 187, 190, 204, 209, 210, 220, 228, 231, 238, 266, 276, 299, 308, 312, 315, 322, 325, 330, 345, 378, 414, 420, 429, 435, 440, 442, 450, 459, 460, 468, 476, 483, 493, 496, 510, 527, 551, 558, 561, 570

Offset: 1

Labos Elemer, Nov 13 2000

Original name was: Numbers n such that gcd(2*n,C(2*n,n))=2*n.
For n a prime power (see A000961) we have gcd(2*n,C(2*n,n))=2. - Arkadiusz Wesolowski, Jul 01 2012
Also, positions where A075055 differs from A000984. - Ralf Stephan, Dec 11 2004
From Peter Bala, Aug 21 2025: (Start)
Also numbers k such that (2*k - 2)!/(k!)^2 is an integer (since (2*k - 1)!/(k!)^2  + (2*k - 2)!/(k!)^2 = 2*Catalan(k-1) for k >= 1). Equivalently, numbers k such that Catalan(k-1) is divisible by k.
Since for prime p, Catalan(p-1) == -1 (mod p), the entries in this list are all nonprime. (End)

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000108, A000961, A001405, A002503, A075055, A000984, A067348.

q:= k-> is(denom((2*k-1)!/(k!)^2)=1):
select(q, [$1..600])[];  # Alois P. Heinz, Feb 06 2025

Select[Range[500], IntegerQ[(2 # - 1)!/#!^2] &] (* Arkadiusz Wesolowski, Jul 01 2012 *)

Appears to be A067348(n)/2. - Benoit Cloitre, Mar 21 2003

Terms >1 are given by A002503+1. - Benoit Cloitre, Dec 09 2017

Name changed by Arkadiusz Wesolowski, Jul 01 2012

A067315 Numbers k such that binomial(k, floor(k/2)) is not divisible by k.

Original entry on oeis.org

4, 6, 8, 10, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 86, 88, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 134, 136, 138, 140

Offset: 1

Labos Elemer, Jan 14 2002

nonn

All the terms are even. - Amiram Eldar, Aug 24 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001405, A042970, A042996 (complement), A067348.

Select[ Range[ 2, 150, 2 ], Mod[ Binomial[ #, #/2 ], # ]>0& ]

is(k) = binomial(k, k\2) % k > 0; \\ Amiram Eldar, Aug 24 2024

A080394 Numbers k such that binomial(k, floor(k/2)) is divisible by k^2.

Original entry on oeis.org

1, 195, 273, 357, 385, 399, 585, 627, 665, 897, 935, 945, 957, 975, 1071, 1085, 1155, 1209, 1235, 1395, 1547, 1581, 1595, 1705, 1771, 1848, 1881, 1925, 1935, 1995, 2035, 2091, 2193, 2255, 2295, 2331, 2365, 2405, 2475, 2574, 2583, 2585, 2639, 2665, 2679

Offset: 1

Labos Elemer, Mar 18 2003

nonn

			Very few values are even, like 1848 and 2574 (in A067348).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001405, A000984, A067348.

Cf. A073076.

Do[s=Binomial[n, Floor[n/2]]/n^2; If[IntegerQ[s], Print[n]], {n, 1, 10000}]

is(k) = !(binomial(k, k\2) % n^2); \\ Amiram Eldar, Aug 11 2024

A048618 Even numbers n such that binomial(n,n/2) is divisible by n/2.

Original entry on oeis.org

2, 4, 12, 30, 40, 56, 84, 90, 132, 154, 176, 182, 208, 220, 252, 280, 306, 312, 340, 374, 380, 408, 418, 420, 440, 456, 462, 476, 480, 532, 552, 598, 616, 624, 630, 644, 650, 660, 690, 736, 756, 828, 840, 858, 870, 880, 884, 900, 918, 920, 928, 936, 952, 966

Offset: 1

Labos Elemer

nonn

			For n=30, binomial(30,15) = 155117520 = 15^10341168, so 30 is a term.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
M. Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems, sect. III: Binomial coefficients modulo integers, binomod.gp (V. 1.4, 11/2015).

Cf. A001405, A020475, A014847, A067348 (binomial(2*n,n) is divisible by 2*n).

a:=[];
for n from 1 to 1000 do if ( binomial(2*n,n) mod n ) = 0 then a:=[op(a),2*n]; fi; od;
a;   # N. J. A. Sloane, Aug 03 2017

Select[Range[2,1000,2],Mod[Binomial[#,#/2],#/2]==0&] (* Harvey P. Dale, Jan 23 2025 *)

a(n) = 2 * A014847(n). - Rémy Sigrist, Aug 27 2017

Definition corrected by N. J. A. Sloane, Aug 03 2017

A080392 Numbers k such that A000984(k) mod k = 0 and A080383(k) != 7.

Original entry on oeis.org

2, 420, 920, 1122, 1218, 1892, 1978, 2444, 2914, 3198, 3782, 4028, 4136, 4292, 4664, 4958, 4960, 5330, 5762, 5986, 6020, 6032, 6710, 6834, 6864, 6882, 6954, 6956, 6968, 7106, 7130, 7140, 7238, 7254, 7448, 7616, 8178, 8190, 8400, 8692, 9462, 9506, 10712, 11060, 11288

Offset: 1

Labos Elemer, Mar 17 2003

nonn

Numbers arising in A067348 and not present in A080385.

Even numbers n such that n divides binomial(n, [n/2]) and A010551(n) does not divide j!*(n-j)! exactly 7 times for j = 0..n. - Peter Luschny, Aug 04 2017

			A080383(2) = 3;
A080383(420) = 11;
A080383(920) = 11;
A080383(1122) = 9;
A080383(1218) = 9.

David A. Corneth, Table of n, a(n) for n = 1..274 (Terms <= 60000)

Cf. A000984, A001405, A067348, A080383, A080385.

isa := proc(n)  local bn, bm;
if n mod 2 = 0 then bn := binomial(n, iquo(n,2)):
if modp(bn, n) = 0 then
   bm := (n, j) -> `if`(modp(bn, binomial(n, j)) = 0, 1, 0):
   return 1 <> add(bm(n, j), j=2..iquo(n,2)-1)
fi fi; false end:
select(isa, [$1..5000]); # Peter Luschny, Aug 04 2017

Do[s=Count[Table[IntegerQ[Binomial[n, Floor[n/2]]/ Binomial[n, j]], {j, 0, n}], True]; s1=IntegerQ[Binomial[n, n/2]/n]; If[ !Equal[s, 7] && Equal[s1, True], Print[n]], {n, 1, 10000}]
(* Second program: *)
Select[Range@ 5000, Function[n, And[Divisible[Binomial[n, n/2], n], Count[Table[Divisible[Binomial[n, Floor[n/2]], Binomial[n, j]], {j, 0, n}], True] != 7]]] (* Michael De Vlieger, Jul 30 2017 *)

More terms from Michael De Vlieger, Jul 30 2017

A080395 Even numbers k such that the central binomial coefficient A000984(k, k/2) is divisible by k^2.

Original entry on oeis.org

1848, 2574, 4004, 4290, 6732, 7480, 8398, 12012, 12236, 17710, 20930, 22770, 24570, 24650, 24882, 25080, 25194, 26796, 27132, 30160, 31668, 36540, 36708, 37674, 37944, 38454, 47124, 47740, 51282, 51480, 53200, 57288, 62160, 68376, 69930, 70840, 73260, 75480, 83640

Offset: 1

Labos Elemer, Mar 18 2003

nonn

a(n)/2 is a term of A121943 for all n. - Amiram Eldar, Mar 07 2022

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001405, A000984, A067348, A080394, A121943.

Do[s=Binomial[n, n/2]/n^2; If[IntegerQ[s], Print[n]], {n, 1, 50000}]
Select[2Range[50000],Mod[Binomial[#,#/2],#^2]==0&] (* Harvey P. Dale, Jan 27 2025 *)

Name corrected and more terms added by Amiram Eldar, Mar 07 2022

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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A002503 Numbers k such that binomial(2*k,k) is divisible by (k+1)^2.

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A058008 Numbers k such that (2*k - 1)!/(k!)^2 is an integer.

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A067315 Numbers k such that binomial(k, floor(k/2)) is not divisible by k.

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A080394 Numbers k such that binomial(k, floor(k/2)) is divisible by k^2.

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A048618 Even numbers n such that binomial(n,n/2) is divisible by n/2.

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A080392 Numbers k such that A000984(k) mod k = 0 and A080383(k) != 7.

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