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A014847
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Numbers k such that k-th Catalan number C(2k,k)/(k+1) is divisible by k.
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40
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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
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OFFSET
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1,2
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COMMENTS
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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
The asymptotic density of this sequence is 0.11424743... (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021
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LINKS
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FORMULA
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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
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MAPLE
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filter:= proc(n) local F, f, r, c, t, j;
F:= ifactors(n)[2];
for f in F do
r:= convert(n, base, f[1]);
c:= 0: t:= 0:
for j from 1 to nops(r) do
if 2*r[j]+c >= f[1] then
c:= 1; t:= t+1;
else c:= 0
fi;
od;
if t < f[2] then return false fi;
od;
true
end proc:
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MATHEMATICA
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fQ[n_] := IntegerQ[Binomial[2n, n]/ n]; Select[ Range@495, fQ@# &] (* Robert G. Wilson v, Jun 19 2006 *)
Select[Table[{CatalanNumber[n], n}, {n, 500}], Divisible[#[[1]], #[[2]]]&][[All, 2]] (* Harvey P. Dale, Nov 07 2022 *)
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PROG
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(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
for n in range(1, 10**3):
if not b % n:
(Magma) [n: n in [1..500] | IsZero((Binomial(2*n, n) div (n+1)) mod n)]; // Vincenzo Librandi, Jan 29 2016
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CROSSREFS
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Cf. A000108, A000984, A120622, A120623, A120624, A120625, A120626, A121943, A282163, A282346, A283073, A283074, A282672.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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