A002503 -id:A002503 - cp's OEIS frontend

A065351 Duplicate of A002503.

5, 14, 27, 41, 44, 65, 76, 90, 109, 125, 139, 152, 155, 169, 186, 189, 203

Offset: 0

dead

A176358 Partial sums of A002503.

5, 19, 46, 87, 131, 196, 272, 362, 471, 596, 735, 887, 1042, 1211, 1397, 1586, 1789, 1997, 2206, 2425, 2652, 2882, 3119, 3384, 3659, 3957, 4264, 4575, 4889, 5210, 5534, 5863, 6207, 6584, 6997, 7416, 7844, 8278, 8717, 9158, 9607, 10065, 10524, 10991

Offset: 1

Jonathan Vos Post, Apr 15 2010

			a(4) = 5 + 14 + 27 + 41 = 87.

Cf. A002503.

Accumulate[Select[Range[800],Divisible[Binomial[2#,#],(#+1)^2]&]]  (* Harvey P. Dale, Apr 18 2011 *)

a(n) = Sum_{i=1..n} A002503(i).

Corrected by Harvey P. Dale, Apr 18 2011

A067348 Even numbers n such that binomial(n, [n/2]) is divisible by n.

Original entry on oeis.org

2, 12, 30, 56, 84, 90, 132, 154, 182, 220, 252, 280, 306, 312, 340, 374, 380, 408, 418, 420, 440, 456, 462, 476, 532, 552, 598, 616, 624, 630, 644, 650, 660, 690, 756, 828, 840, 858, 870, 880, 884, 900, 918, 920, 936, 952, 966, 986, 992, 1020, 1054, 1102

Offset: 1

Dean Hickerson, Jan 16 2002

nonn

This sequence has a surprisingly large overlap with A080385(n); a few values, 2, 420, 920 are exceptional. This means that usually A080383(A067348(n))=7. - Labos Elemer, Mar 17 2003

Conjecture: sequence contains most of 2*A000384(k). Exceptions are k = 8, 18, 20, 23, 35, ... - Ralf Stephan, Mar 15 2004

David A. Corneth, Table of n, a(n) for n = 1..12000

Subsequence of A042996.

Cf. A000984, A067315, A080385.

Select[Range[2, 1200, 2], Mod[Binomial[ #, #/2], # ]==0&]

val(n, p) = my(r=0); while(n, r+=n\=p);r
is(n) = {if(valuation(n, 2) == 0, return(0)); my(f = factor(n)); for(i=1, #f~, if(val(n, f[i, 1]) - 2 * val(n/2, f[i, 1]) - f[i, 2] < 0, return(0))); return(1)} \\ David A. Corneth, Jul 29 2017

a(n) = 2*A002503(n-2) + 2.

Appears to be 2*A058008(n). - Benoit Cloitre, Mar 21 2003

Name clarified by Peter Luschny, Aug 04 2017

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

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

A065351 Duplicate of A002503.

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A176358 Partial sums of A002503.

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A067348 Even numbers n such that binomial(n, [n/2]) is divisible by n.

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

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

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

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Mathematica

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Extensions