OFFSET
1,2
COMMENTS
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
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..2000
FORMULA
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
EXAMPLE
a(4) = 80 = 8 * 10, as 7 is the 4th prime and 11 is the 5th prime.
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
MAPLE
f:= proc(n) local p, q;
p:= ithprime(n); q:= ithprime(n+1);
2^((q-p)/2)*floor(q/2)!/floor(p/2)!
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Aug 28 2016
MATHEMATICA
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 *)
PROG
(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", "))
(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
(PARI) a(n) = {my(p1 = prime(n), p2 = nextprime(p1 + 1));
2^((p2-p1)\2) * prod(i=(p1+1)\2, (p2-1)\2, i)} \\ David A. Corneth, Aug 21 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Apr 22 2001
EXTENSIONS
Corrected and extended by Ralf Stephan, Mar 22 2003
Name simplified by David A. Corneth, Aug 21 2016
STATUS
approved