A014847 -id:A014847 - cp's OEIS frontend

A235037 Number of terms of A014847 that are not greater than n.

1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Bruno Berselli, Jan 05 2014 - sequence suggested by Umberto Cerruti (University of Turin, Italy)

nonn

			a(6)=3 because the terms of A014847 not greater than 6 are 1,2,6.
a(17)=4 because the terms of A014847 not greater than 17 are 1,2,6,15.

Bruno Berselli, Table of n, a(n) for n = 1..2000

Cf. A014847.

a[n_] := Module[{ris}, ris = {}; Do[If[Mod[Binomial[2 k, k], k] == 0, AppendTo[ris, k]], {k, n}]; Length[ris]]; Table[a[n], {n, 100}]

A059288 a(n) = binomial(2*n,n) mod n.

Original entry on oeis.org

0, 0, 2, 2, 2, 0, 2, 6, 2, 6, 2, 4, 2, 6, 0, 6, 2, 6, 2, 0, 6, 6, 2, 12, 2, 6, 20, 0, 2, 4, 2, 6, 9, 6, 7, 16, 2, 6, 20, 20, 2, 0, 2, 4, 0, 6, 2, 12, 2, 6, 3, 44, 2, 6, 32, 32, 39, 6, 2, 36, 2, 6, 12, 6, 5, 0, 2, 36, 66, 40, 2, 36, 2, 6, 45, 32, 0, 66, 2, 20, 20, 6, 2

Offset: 1

N. J. A. Sloane, Jan 25 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..5000 from T. D. Noe)

Cf. A014847, A053214, A059289, A000108.

binomial(2*n,n) mod n;
seq(irem(binomial(2*n,n),n),n=1..83); # Zerinvary Lajos, Apr 20 2008

Table[Mod[Binomial[2*n, n], n], {n, 1, 25}] (* G. C. Greubel, Jan 04 2017 *)

a(n) = binomial(2*n, n) % n; \\ Harry J. Smith, Jun 25 2009

a(n) = Catalan(n) mod n. - Jonathan Sondow, Dec 13 2013

a(p) = 2, p an odd prime (provable using Wolstenholme's theorem). - David Trimas, Feb 11 2025

A121943 Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2.

Original entry on oeis.org

1, 924, 1287, 2002, 2145, 3366, 3640, 3740, 4199, 6006, 6118, 6552, 7480, 7920, 8580, 8855, 10465, 10920, 11385, 11592, 12285, 12325, 12441, 12540, 12597, 12920, 13224, 13398, 13566, 15080, 15834, 18270, 18354, 18837, 18972, 19227, 23562, 23870, 25641, 25740

Offset: 1

Tanya Khovanova, Sep 03 2006

nonn

Equivalently, numbers n such that the n-th Catalan number C(2n,n)/(n+1) is divisible by n^2. - Lucian Craciun, Feb 09 2017

The asymptotic density of this sequence is 0.00322778... (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020.

Cf. A000108, A000984, A014847, A282163, A282346, A283073, A283074, A282672.

Select[Table[n, {n, 20000}], IntegerQ[Binomial[2#, # ]/#^2] &]

lista(nn) = {for(n=1, nn, if(Mod(binomial(2*n, n), n^2) == 0, print1(n, ", ")));} \\ Altug Alkan, Mar 27 2016

from _future_ import division
A121943_list, b = [], 2
for n in range(1,10**5):
    if not b % (n**2):
        A121943_list.append(n)
    b = b*(4*n+2)//(n+1) # Chai Wah Wu, Mar 27 2016

A282163 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^3.

Original entry on oeis.org

1, 154836, 985320, 1108536, 1113959, 1492260, 1576696, 1632708, 1649238, 1684540, 1805570, 1988008, 2508792, 2548810, 2550408, 2659260, 2698740, 2746590, 2995122, 3074552, 3286710, 3330795, 3538458, 3574200, 3730155, 4039932, 4160240, 4318548, 4374370, 4426695, 4523985

Offset: 1

Lucian Craciun, Feb 07 2017

nonn

Equivalently, numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^3. - Lucian Craciun, Feb 09 2017

The asymptotic density of this sequence is 0.000031511777... (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

			The central binomial coefficient C(2*154836,154836) is divisible by 154836^3.

Lucian Craciun, Table of n, a(n) for n = 1..16000 (corrected and extended by Giovanni Resta)
Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020.
Wikipedia Mathematics Reference Desk, n^4 Divides Central Binomial Coefficient.

Cf. A000108, A000984, A014847, A121943, A282346, A283073, A283074, A282672.

A282163 := proc (n, m) local a, cbc, k; a := {}; cbc := binomial(2*n, n); for k from n+1 to m do cbc := cbc*(4-2/k); if type(cbc/k^3, integer) then a := `union`(a, {k}) end if end do; a end proc; A282163(0, 10^6)

Select[Table[n, {n, 10^6}], IntegerQ[Binomial[2#, #]/#^3] &] (* for small n *)
n := 0; m := 10^6; A282163 := {}; cbc := Binomial[2n, n]; For[k := n+1, k <= m, k++, {cbc *= 4-2/k, If[IntegerQ[cbc/k^3], A282163 = Append[A282163, k]]}] (* for large m *)
A282163:={}; k:=3; For[n:=1, n<=10^6, n++, {f=FactorInteger[n], For[j:=1, j<=Length[f], j++, {b=True, If[Sum[Floor[2n/f[[j, 1]]^i]-2 Floor[n/f[[j, 1]]^i], {i, 1, Length[IntegerDigits[2n, f[[j, 1]]]]}]A282163=Append[A282163, n]]}] (* Legendre's formula for drastic time reduction, Lucian Craciun, Feb 28 2017; optimized by Lucian Craciun, Mar 02 2017 *)

A282672 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^6.

Original entry on oeis.org

1, 1138842118714300, 1605078397568386, 1785922862964240, 1878157384495600, 2020105305316098, 2055406015517400, 2071857393746595, 2310442996851990, 2450253379658700, 2513216312053944, 2966830431558840, 2990886595291870, 3228082757486928, 3318987930069240

Offset: 1

Giovanni Resta, Mar 16 2017

nonn

Also numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^6.

The asymptotic density of this sequence is 3.40390904801... *10^(-13) (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

			Let E(n,p) be the exponent of the prime p in the factorization of n. Note that E(n!,p) can be easily found with Legendre's formula without computing n!. Then, t = 1138842118714300 is in the sequence because for each prime p dividing t we have E(C(2*t,t),p) = E((2*t)!,p) - 2*E(t!,p) >= 6*E(t,p).

Giovanni Resta, Table of n, a(n) for n = 1..97
Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020.

Cf. A000108, A000984, A014847, A121943, A282163, A282346, A283073, A283074.

A283073 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^4.

Original entry on oeis.org

1, 227736432, 338956200, 386160984, 482213160, 544508118, 548823405, 715592220, 726922482, 731987190, 1427877360, 1448431600, 1467104760, 1490842353, 1491241258, 1504640335, 1646570115, 1852712100, 1923506200, 1923927460, 1924947570, 2056580995, 2064409413

Offset: 1

Lucian Craciun, Feb 28 2017

nonn

Equivalently, numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^4.

The asymptotic density of this sequence is 1.330129946... * 10^(-7) (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

			The central binomial coefficient C(2*227736432,227736432) is divisible by 227736432^4.

Giovanni Resta, Table of n, a(n) for n = 1..1002
Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020.
Wikipedia Mathematics Reference Desk, n^4 Divides Central Binomial Coefficient.

Cf. A000108, A000984, A014847, A121943, A282163, A282346, A283074, A282672.

A283073:={}; k:=4; For[n:=1, n<=10^9, n++, {f=FactorInteger[n], For[j:=1, j<=Length[f], j++, {b=True, If[Sum[Floor[2n/f[[j, 1]]^i]-2 Floor[n/f[[j, 1]]^i], {i, 1, Length[IntegerDigits[2n, f[[j, 1]]]]}]A283073=Append[A283073, n]]}] (* Legendre's formula for drastic time reduction *)

a(11)-a(22) from Giovanni Resta, Feb 28 2017

A283074 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^5.

Original entry on oeis.org

1, 84331608790, 94482127740, 164273806200, 438726722148, 541278246600, 549361342530, 808172086449, 912226745430, 959218287720, 1017676553985, 1017868271175, 1078659050256, 1286556180525, 1418394308100, 1475851476960, 1489765799610, 1535790227400, 1562434592400, 1642639268270

Offset: 1

Lucian Craciun, Feb 28 2017

nonn

Equivalently, numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^5.

The asymptotic density of this sequence is 2.83248121476... * 10^(-10) (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

			The central binomial coefficient C(2*84331608790,84331608790) is divisible by 84331608790^5.

Giovanni Resta, Table of n, a(n) for n = 1..123 (terms < 10^13).
Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020.

Cf. A000108, A000984, A014847, A121943, A282163, A282346, A283073, A282672.

a(3)-a(20) from Giovanni Resta, Mar 03 2017

A282346 Least number m > 1 such that the central binomial coefficient C(2m,m) is divisible by m^n.

Original entry on oeis.org

2, 924, 154836, 227736432, 84331608790, 1138842118714300

Offset: 1

Lucian Craciun and Robert G. Wilson v, Feb 12 2017

Equivalently, least number m > 1 such that the m-th Catalan number C(2m,m)/(m+1) is divisible by m^n. - Lucian Craciun, Mar 01 2017
a(6) <= 4380346834858680. - David A. Corneth, Mar 04 2017
a(7) <= 2404760413443713325. - Giovanni Resta, Mar 16 2017

Wikipedia, Kummer's theorem.

Cf. A000108, A000984, A014847, A121943, A282163, A283073, A283074, A282672.

f[n_] := Block[{k = 2}, While[Mod[Binomial[2k, k], k^n] > 0, k++]; k]

a(4)-a(5) from Giovanni Resta, Feb 23 2017

a(6) from Giovanni Resta, Mar 15 2017

A004782 Numbers k such that 2(2k-3)!/(k!(k-1)!) is an integer.

Original entry on oeis.org

2, 3, 7, 16, 21, 29, 43, 46, 67, 78, 89, 92, 105, 111, 127, 141, 154, 157, 171, 188, 191, 205, 210, 211, 221, 229, 232, 239, 241, 267, 277, 300, 309, 313, 316, 323, 326, 331, 346, 369, 379, 415, 421, 430, 436, 441, 443, 451, 460, 461, 465, 469, 477

Offset: 1

R. K. Guy

nonn

Superset of A081767, as proved by Luke Pebody. Terms not in A081767 include 3, 7, 127, 511, ... - Ralf Stephan, Oct 12 2004
See A260642 for A004782 \ A081767. - M. F. Hasler, Nov 11 2015
Equivalently, numbers k such that binomial(2k-3,k-1) == 0 (mod k*(k-1)/2), or: binomial(2k-2,k-1) == 0 (mod k^2-k), or: the Catalan number A000108(k-1) is divisible by k-1, i.e., a(n) = A014847(n) + 1. Indeed, 2(2k-3)!/(k!*(k-1)!) = 2(2k-2)!/(k!(k-1)!(2k-2)) = C(k-1)/(k-1). - M. F. Hasler, Nov 11 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Select[Range[500], IntegerQ[2 (2 # - 3)!/(#! (# - 1)!)] &] (* Arkadiusz Wesolowski, Sep 06 2011 *)

for(n=2, 999, binomial(2*n-2, n-1)%(n^2-n)||print1(n", "))

is_A004782(n)=!binomod(2*n-2, n-1, n^2-n) \\ Using http://home.gwu.edu/~maxal/gpscripts/binomod.gp by M. Alekseyev. - M. F. Hasler, Nov 11 2015

a(n) = A014847(n) + 1. - Enrique Pérez Herrero, Feb 03 2013

Offset corrected and initial term added by Arkadiusz Wesolowski, Sep 06 2011

A071416 a(n) = gcd(n, binomial(2*n, n)).

Original entry on oeis.org

1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 15, 2, 1, 6, 1, 20, 3, 2, 1, 12, 1, 2, 1, 28, 1, 2, 1, 2, 3, 2, 7, 4, 1, 2, 1, 20, 1, 42, 1, 4, 45, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 8, 3, 2, 1, 12, 1, 2, 3, 2, 5, 66, 1, 4, 3, 10, 1, 36, 1, 2, 15, 4, 77, 6, 1, 20, 1, 2, 1, 28, 5, 2, 3, 88, 1, 10, 91, 4, 1, 2, 5

Offset: 1

Reinhard Zumkeller, May 29 2002

a(n) = n for n in A014847. - Reinhard Zumkeller

a(n) = gcd(n, C(n)), C(n) the Catalan numbers. - Peter Luschny Oct 06 2011

			a(10) = gcd(10, binomial(20, 10)) = gcd(10, 184756) = 2.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Central Binomial Coefficient

Cf. A000984, A000108.

Table[GCD[n,Binomial[2n,n]],{n,100}] (* Harvey P. Dale, Nov 10 2011 *)

cp's OEIS Frontend

A235037 Number of terms of A014847 that are not greater than n.

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A059288 a(n) = binomial(2*n,n) mod n.

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A121943 Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2.

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A282163 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^3.

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A282672 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^6.

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A283073 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^4.

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A283074 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^5.

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A282346 Least number m > 1 such that the central binomial coefficient C(2m,m) is divisible by m^n.

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A004782 Numbers k such that 2(2k-3)!/(k!(k-1)!) is an integer.