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A061216 a(n) = product of all even numbers between n-th prime and (n+1)-st prime.

%I A061216 #37 Jan 07 2025 05:40:12
%S A061216 1,4,6,80,12,224,18,440,17472,30,39168,1520,42,2024,124800,175392,60,
%T A061216 261888,4760,72,438672,6560,635712,74718720,9800,102,11024,108,12320,
%U A061216 356925975275520,16640,2405568,138,61857653760,150,3651648,4095360
%N A061216 a(n) = product of all even numbers between n-th prime and (n+1)-st prime.
%C A061216 Previous name used "even composite numbers", but if an even number is strictly between two primes, it is composite. So the word 'composite' isn't needed in the title. - _David A. Corneth_, Aug 21 2016
%H A061216 Harry J. Smith, <a href="/A061216/b061216.txt">Table of n, a(n) for n = 1..2000</a>
%F A061216 a(n) = 2^((prime(n+1)-prime(n))/2) * ((prime(n+1)-1)/2)!/((prime(n)-1)/2)! for n >= 2. - _Robert Israel_, Aug 28 2016
%e A061216 a(4) = 80 = 8 * 10, as 7 is the 4th prime and 11 is the 5th prime.
%e A061216 a(9) = 17472. Let p_(n) = prime(n). p_(9) = 23, p_(10) = 29. The number of even numbers between 23 and 29 is floor((29 - 23) / 2) = 3. So a(9) is 2^3 * (23 + 1)/2 * ... * (29 - 1)/2 = 17472. - _David A. Corneth_, Aug 21 2016
%p A061216 f:= proc(n) local p,q;
%p A061216   p:= ithprime(n); q:= ithprime(n+1);
%p A061216   2^((q-p)/2)*floor(q/2)!/floor(p/2)!
%p A061216 end proc:
%p A061216 f(1):= 1:
%p A061216 map(f, [$1..100]); # _Robert Israel_, Aug 28 2016
%t A061216 f[n_]:=Module[{pn=Prime[n],pn1=Prime[n+1]},Times@@Range[pn+1,pn1,2]]; Table[f[i], {i, 45}] (* _Harvey P. Dale_, Jan 16 2011 *)
%o A061216 (PARI) for(n=1,50,p=1;for(k=prime(n)+1, prime(n+1)-1,if(k%2==0,p=p*k));print1(p","))
%o A061216 (PARI) n=0; q=2; forprime (p=3, prime(2001), a=1; for (i=q + 1, p - 1, if (i%2==0, a*=i)); q=p; write("b061216.txt", n++, " ", a) ) \\ _Harry J. Smith_, Jul 19 2009
%o A061216 (PARI) a(n) = {my(p1 = prime(n), p2 = nextprime(p1 + 1));
%o A061216 2^((p2-p1)\2) * prod(i=(p1+1)\2,(p2-1)\2,i)} \\ _David A. Corneth_, Aug 21 2016
%Y A061216 Cf. A061214, A061215.
%K A061216 nonn
%O A061216 1,2
%A A061216 _Amarnath Murthy_, Apr 22 2001
%E A061216 Corrected and extended by _Ralf Stephan_, Mar 22 2003
%E A061216 Name simplified by _David A. Corneth_, Aug 21 2016