The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A014847 Numbers k such that k-th Catalan number C(2k,k)/(k+1) is divisible by k. 22
 1, 2, 6, 15, 20, 28, 42, 45, 66, 77, 88, 91, 104, 110, 126, 140, 153, 156, 170, 187, 190, 204, 209, 210, 220, 228, 231, 238, 240, 266, 276, 299, 308, 312, 315, 322, 325, 330, 345, 368, 378, 414, 420, 429, 435, 440, 442, 450, 459, 460, 464, 468, 476, 483, 493 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The sequence does not contain any odd primes p (follows by quadratic reciprocity and field structure of Z/pZ). Aside from the first 2 terms, all other terms are composite integers. - Thomas M. Bridge, Nov 03 2013 Equivalently, numbers such that binomial(2n, n) = 0 (mod n). Indices of zeros in A059288. See A260640 (and A260636) for the analogs for 3n. - M. F. Hasler, Nov 11 2015 The 2nd comment is true because gcd(n,n+1) = 1 and n+1 divides C(2n,n). The 1st comment then follows, because prime p does not divide C(2p,p) = 2p*(2p-1)*...*(p+1)/(p*(p-1)*...*1) unless p = 2. - Jonathan Sondow, Jan 07 2018 A number n is in the sequence if and only if, for each prime p dividing n, the number of carries in the addition n+n in base p is at least the p-adic valuation of n. In particular, if n is squarefree, the condition is that at least one base-p digit of n is at least p/2. - Robert Israel, Jan 07 2018 If A is the set of all a(k)'s, Pomerance proved that the upper density of A is at most 1 - log 2 = 0.30685... and conjectured that A has positive lower density. I improved Pomerance's result by showing that the upper density of A is at most 1 - log 2 - 0.05551 = 0.25134... Numerically, this upper density seems to be less than 0.11. - Carlo Sanna, Jan 28 2018 LINKS Franklin T. Adams-Watters and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n=1..1069 (a(n) <= 10000) from Franklin T. Adams-Watters Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems, sect. III: Binomial coefficients modulo integers, binomod.gp (V. 1.4, 11/2015). Christian Ballot, Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions, J. Int. Seq., Vol. 21 (2018), Article 18.6.5. Kevin Ford, Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), arXiv:1909.03903 [math.NT], 2019. Carl Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644. Carlo Sanna, Central binomial coefficients divisible by or coprime to their indices, Int. J. Number Theory (2018). Eric Weisstein's World of Mathematics, Disk Line Picking FORMULA It seems that a(n)/n is bounded and more precisely that lim_{n -> infinity} a(n)/n = C exists with 9 <= c < 10. - Benoit Cloitre, Aug 13 2002 a(n) = A004782(n) - 1. - Enrique Pérez Herrero, Feb 03 2013 MAPLE filter:= proc(n) local F, f, r, c, t, j;   F:= ifactors(n);   for f in F do     r:= convert(n, base, f);     c:= 0: t:= 0:     for j from 1 to nops(r) do       if 2*r[j]+c >= f then           c:= 1; t:= t+1;       else c:= 0       fi;     od;     if t < f then return false fi;   od;   true end proc: select(filter, [\$1..1000]); # Robert Israel, Jan 07 2018 MATHEMATICA fQ[n_] := IntegerQ[Binomial[2n, n]/ n]; Select[ Range@495, fQ@# &] (* Robert G. Wilson v, Jun 19 2006 *) PROG (PARI) is_A014847(n)=!binomod(2*n, n, n) \\ Suitable for large n. Using binomod.gp by M. Alekseyev, cf. links. - M. F. Hasler, Nov 11 2015 (PARI) for(n=1, 1e3, if(binomial(2*n, n)/(n+1) % n==0, print1(n", "))) \\ Altug Alkan, Nov 11 2015 (Python) from __future__ import division A014847_list, b = [], 1 for n in range(1, 10**3):     if not b % n:         A014847_list.append(n)     b = b*(4*n+2)//(n+2) # Chai Wah Wu, Jan 27 2016 (MAGMA) [n: n in [1..500] | IsZero((Binomial(2*n, n) div (n+1)) mod n)]; // Vincenzo Librandi, Jan 29 2016 CROSSREFS Cf. A000108, A000984, A120622, A120623, A120624, A120625, A120626, A121943, A282163, A282346, A283073, A283074, A282672. Sequence in context: A294942 A227307 A129631 * A013636 A144653 A276782 Adjacent sequences:  A014844 A014845 A014846 * A014848 A014849 A014850 KEYWORD nonn AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 17 20:36 EST 2020. Contains 330987 sequences. (Running on oeis4.)