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

1,2

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

Cf. A061214, A061215.

Sequence in context: A013128 A012971 A197315 * A058163 A303211 A264374

Adjacent sequences: A061213 A061214 A061215 * A061217 A061218 A061219

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