A065344 -id:A065344 - cp's OEIS frontend

A065347 Positions of zeros in A065344, i.e., binomial(2n,n) mod ((n+1)*(n+2)) = 0.

3, 5, 8, 9, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 32, 33, 35, 36, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90

Offset: 1

Labos Elemer, Oct 30 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000
C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644.

Cf. A000108, A065344, A065345, A065346, A065348, A065349.

ris = {}; Do[If[Mod[Binomial[2 n, n], (n + 1) (n + 2)] == 0, AppendTo[ris, n]], {n, 100}]; ris (* Bruno Berselli, Jan 06 2014 *)

isok(k) = { binomial(2*k, k) % ((k + 1)*(k + 2)) == 0 } \\ Harry J. Smith, Oct 17 2009

A065349 Positions of zeros in A065346.

Original entry on oeis.org

19, 41, 42, 43, 49, 50, 53, 54, 55, 58, 67, 71, 74, 75, 95, 97, 98, 99, 102, 103, 123, 135, 138, 139, 145, 149, 153, 154, 155, 159, 163, 165, 166, 167, 168, 169, 170, 173, 174, 175, 178, 179, 180, 183, 184, 185, 191, 193, 194, 195, 197, 198, 199, 200, 201, 202

Offset: 1

Labos Elemer, Oct 30 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000
C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644.

Cf. A000108, A065344, A065345, A065346, A065347, A065348.

ris = {}; Do[If[Mod[Binomial[2 n, n], (n + 1) (n + 2) (n + 3) (n + 4)] == 0, AppendTo[ris, n]], {n, 250}]; ris (* Bruno Berselli, Jan 06 2014 *)

isok(k) = { binomial(2*k, k) % ((k + 1)*(k + 2)*(k + 3)*(k + 4)) == 0 } \\ Harry J. Smith, Oct 17 2009

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

A065345 a(n) = binomial(2n,n) mod ((n+1)(n+2)*(n+3)).

Original entry on oeis.org

2, 6, 20, 70, 252, 420, 552, 0, 1100, 1144, 0, 1456, 1400, 2040, 2448, 0, 3420, 0, 0, 0, 6072, 5520, 0, 5850, 13104, 0, 12992, 17980, 22320, 27280, 5984, 7854, 7140, 15540, 0, 36556, 13832, 0, 45920, 24682, 0, 0, 0, 0, 51888, 0, 23520, 0, 0, 0, 0, 94446, 0, 0, 0

Offset: 1

Labos Elemer, Oct 30 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000
C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644.

Cf. A000108, A065344, A065346, A065347, A065348, A065349.

Table[Mod[Binomial[2 n, n], (n + 1) (n + 2) (n + 3)], {n, 60}] (* Harvey P. Dale, Feb 21 2012 *)

a(n) = { binomial(2*n, n) % ((n + 1)*(n + 2)*(n + 3)) } \\ Harry J. Smith, Oct 17 2009

A065346 a(n) = binomial(2n, n) mod ((n+1)(n+2)(n+3)(n+4)).

Original entry on oeis.org

2, 6, 20, 70, 252, 924, 3432, 990, 14300, 16588, 17472, 39676, 4760, 18360, 46512, 29070, 30780, 87780, 0, 191268, 273240, 322920, 140400, 58500, 190008, 350784, 402752, 611320, 81840, 649264, 41888, 164934, 264180, 295260, 1316016, 694564

Offset: 1

Labos Elemer, Oct 30 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644.

Cf. A000108, A065344, A065345, A065347, A065348, A065349.

Table[Mod[Binomial[2 n, n], (n + 1) (n + 2) ((n + 3) (n + 4))], {n, 100}] (* Bruno Berselli, Jan 06 2014 *)

a(n) = { binomial(2*n, n) % ((n + 1)*(n + 2)*(n + 3)*(n + 4)) } \\ Harry J. Smith, Oct 17 2009

A065348 Positions of zeros in A065345.

Original entry on oeis.org

8, 11, 16, 18, 19, 20, 23, 26, 35, 38, 41, 42, 43, 44, 46, 48, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 64, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 80, 83, 86, 89, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 107, 110, 113, 116, 119, 123, 124, 128, 130, 131, 134

Offset: 1

Labos Elemer, Oct 30 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000
Carl Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644.

Cf. A000108, A065344, A065345, A065346, A065347, A065349.

ris = {}; Do[If[Mod[Binomial[2 n, n], (n + 1) (n + 2) (n + 3)] == 0,
AppendTo[ris, n]], {n, 150}]; ris (* Bruno Berselli, Jan 06 2014 *)

isok(k) = { binomial(2*k, k) % ((k + 1)*(k + 2)*(k + 3)) == 0 } \\ Harry J. Smith, Oct 17 2009

A065350 a(n) = binomial(2*n, n) mod (n+1)^2.

Original entry on oeis.org

2, 6, 4, 20, 0, 42, 40, 72, 20, 110, 120, 156, 56, 0, 208, 272, 108, 342, 200, 378, 176, 506, 432, 600, 260, 459, 0, 812, 840, 930, 928, 396, 476, 490, 360, 1332, 608, 1131, 1200, 1640, 0, 1806, 880, 0, 920, 2162, 864, 2352, 1100, 1224, 208, 2756, 1296, 2145

Offset: 1

Labos Elemer, Oct 30 2001

nonn

a(A002503(n)) = 0. - Reinhard Zumkeller, Sep 16 2014

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000108, A065344-A065349, A002503.

Cf. A002503, A000984, A000290.

a065350 n = a065350_list !! (n-1)
a065350_list = zipWith mod (tail a000984_list) (drop 2 a000290_list)
-- Reinhard Zumkeller, Sep 16 2014

Table[Mod[Binomial[2 n, n], (n + 1)^2], {n, 100}] (* Bruno Berselli, Jan 06 2014 *)

a(n) = { binomial(2*n, n) % (n + 1)^2 } \\ Harry J. Smith, Oct 17 2009

A065352 Smallest m such that C(2m,m) is divisible by (m+n)!/m!.

Original entry on oeis.org

1, 3, 8, 19, 42, 153, 216, 375, 950, 3565, 4068, 12273, 12274, 31729, 122352, 131023, 458222, 522221, 1046508, 3145451, 6291178, 12320745, 16769000, 56623079, 113246182, 267780069, 469745636, 671088611, 1879015394, 2146959329, 6442418144, 16642932703, 16911433694, 60129279965, 206091288540

Offset: 1

Labos Elemer, Oct 31 2001

nonn

For n=1 see Catalan numbers A000108.
Heuristically one can observe that a(n) + n + 1 has a 'high' valuation of 2. For n = 17..25 we have 2^8|(a(n) + n + 1). - David A. Corneth, Mar 28 2021
Since (m+n)!/m! = C(m+n,m) * n!, Kummer's theorem implies that A000120(a(n)) >= A007814(n!) = A011371(n) = n - A000120(n), and a(n) >= 2^(n-1). - Max Alekseyev, Sep 24 2024

			n=4: a(4)=19 means that C(38,19)=35345263800 is divisible by (19+1)(19+2)(19+3)(19+4)=23!/19!=20*21*22*23=215520; the quotient is 166315. Smaller (<19) central binomial coefficients are not divisible by such a product of 4 successive terms; the corresponding quotients for n = 1, 2, 3, 4, 5,... are 1, 1, 13, 166315, 9120910752273999,...

Max Alekseyev, Table of n, a(n) for n = 1..100
David A. Corneth, PARI program

Cf. A000120, A065344-A065350, A002503, A000108, A000984, A007814, A011371.

Do[m = 1; While[Not[Divisible[Binomial[2*m,m],(m+n)!/m!]], m++]; Print[m], {n, 1, 16}] (* Vaclav Kotesovec, Sep 05 2019 *)

PARI
```
\\ See Corneth link
```

C(2m, m)=A*((m+1)(m+2)...(m+n-1)(m+n)); a(n) is the smallest such m belonging to n: a(n)=Min(m; Mod(A000984(m), (m+n)!/m!)=0)

More terms from Naohiro Nomoto, Apr 21 2002
a(16)-a(17) from Vaclav Kotesovec, Sep 06 2019
a(18)-a(25) from David A. Corneth, Mar 28 2021
a(26)-a(31) from David A. Corneth confirmed and terms a(32) onward added by Max Alekseyev, Sep 24 2024

cp's OEIS Frontend

A065347 Positions of zeros in A065344, i.e., binomial(2n,n) mod ((n+1)*(n+2)) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A065349 Positions of zeros in A065346.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A065345 a(n) = binomial(2*n,n) mod ((n+1)*(n+2)*(n+3)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A065346 a(n) = binomial(2*n, n) mod ((n+1)*(n+2)*(n+3)*(n+4)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A065348 Positions of zeros in A065345.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A065350 a(n) = binomial(2*n, n) mod (n+1)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A065352 Smallest m such that C(2m,m) is divisible by (m+n)!/m!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A065345 a(n) = binomial(2n,n) mod ((n+1)(n+2)*(n+3)).

A065346 a(n) = binomial(2n, n) mod ((n+1)(n+2)(n+3)(n+4)).