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A212791 Central binomial coefficient CB(n) purged of all primes exceeding (n+1)/2.

%I A212791 #22 Nov 20 2023 15:31:29
%S A212791 1,1,1,2,2,4,1,2,18,36,6,12,12,24,45,90,10,20,2,4,84,168,14,28,700,
%T A212791 1400,2700,5400,360,720,45,90,2970,5940,11550,23100,23100,46200,90090,
%U A212791 180180,8580,17160,780,1560,70200,140400
%N A212791 Central binomial coefficient CB(n) purged of all primes exceeding (n+1)/2.
%C A212791 A simple insight shows that the prime factors decomposition of CB(n) = binomial(n, floor(n/2)) (i) does not contain any prime factor greater than n, (ii) contains exactly once all primes in the interval ((n+1)/2, n]. Hence, CB(n) is divisible by the product P2(n) of all primes in ((n+1)/2, n]. The relatively small elements of this sequence are a(n) = CB(n)/P2(n). For n > 6, they can be shown to be devoid of any prime factor exceeding n/3.
%H A212791 Stanislav Sykora, <a href="/A212791/b212791.txt">Table of n, a(n) for n = 1..1000</a>
%F A212791 a(n) = C(n, floor(n/2))/Product_{n/2 < prime p <= n} p.
%e A212791 CB(21) = binomial(21,10) = 21!/(10!11!) = 352716 is divisible by all primes in (11,21] to 1st power, i.e., by 13*17*19 = 4199. Hence a(21) = 352716/4199 = 84.
%o A212791 (PARI) {lambda1(n) = result=binomial(n,floor((n+1)\2));forprime(p=1+floor((n+1)\2),n,result=result/p);}
%Y A212791 Cf. A001405, A212792, A000040 (primes).
%K A212791 nonn
%O A212791 1,4
%A A212791 _Stanislav Sykora_, May 27 2012