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

1, 4, 6, 80, 12, 224, 18, 440, 17472, 30, 39168, 1520, 42, 2024, 124800, 175392, 60, 261888, 4760, 72, 438672, 6560, 635712, 74718720, 9800, 102, 11024, 108, 12320, 356925975275520, 16640, 2405568, 138, 61857653760, 150, 3651648, 4095360

Offset: 1

Author

Amarnath Murthy, Apr 22 2001

Keywords

• nonn

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

Examples

			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
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A061214, A061215.

Programs

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 *)

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

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

Extensions

Corrected and extended by Ralf Stephan, Mar 22 2003

Name simplified by David A. Corneth, Aug 21 2016