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

%I A065352 #32 Sep 25 2024 09:55:36
%S A065352 1,3,8,19,42,153,216,375,950,3565,4068,12273,12274,31729,122352,
%T A065352 131023,458222,522221,1046508,3145451,6291178,12320745,16769000,
%U A065352 56623079,113246182,267780069,469745636,671088611,1879015394,2146959329,6442418144,16642932703,16911433694,60129279965,206091288540
%N A065352 Smallest m such that C(2m,m) is divisible by (m+n)!/m!.
%C A065352 For n=1 see Catalan numbers A000108.
%C A065352 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
%C A065352 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
%H A065352 Max Alekseyev, <a href="/A065352/b065352.txt">Table of n, a(n) for n = 1..100</a>
%H A065352 David A. Corneth, <a href="/A065352/a065352.gp.txt">PARI program</a>
%F A065352 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)
%e A065352 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,...
%t A065352 Do[m = 1; While[Not[Divisible[Binomial[2*m,m],(m+n)!/m!]], m++]; Print[m], {n, 1, 16}] (* _Vaclav Kotesovec_, Sep 05 2019 *)
%o A065352 (PARI) \\ See Corneth link
%Y A065352 Cf. A000120, A065344-A065350, A002503, A000108, A000984, A007814, A011371.
%K A065352 nonn
%O A065352 1,2
%A A065352 _Labos Elemer_, Oct 31 2001
%E A065352 More terms from _Naohiro Nomoto_, Apr 21 2002
%E A065352 a(16)-a(17) from _Vaclav Kotesovec_, Sep 06 2019
%E A065352 a(18)-a(25) from _David A. Corneth_, Mar 28 2021
%E A065352 a(26)-a(31) from _David A. Corneth_ confirmed and terms a(32) onward added by _Max Alekseyev_, Sep 24 2024