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%I #90 Sep 08 2020 16:13:39
%S 1,4,24,192,1728,17280,207360,2903040,43545600,696729600,12541132800,
%T 250822656000,5267275776000,115880067072000,2781121609728000,
%U 69528040243200000,1807729046323200000,48808684250726400000,1366643159020339200000
%N Compositorial numbers: product of first n composite numbers.
%C a(A196415(n)) = A141092(n) * A053767(A196415(n)). - _Reinhard Zumkeller_, Oct 03 2011
%C For n>11, A000142(n) < a(n) < A002110(n). - _Chayim Lowen_, Aug 18 2015
%C For n = {2,3,4}, a(n) is testably a Zumkeller number (A083207). For n > 4, a(n) is of the form 2^e_1 * p_2^e_2 * … * p_m^e_m, where e_m = 1 and e = floor(log_2(p_m)) < e_1. Therefore, 2^e * p_m^e_m is primitive Zumkeler number (A180332). Therefore, 2^e_1 * p_m^e_m is a Zumkeller number. Therefore, a(n) = 2^e_1 * p_m^e_m * r, where r is relatively prime to 2*p_m is a Zumkeller number. Therefore, for n > 1, a(n) is a Zumkeller number (see my proof at A002182 for details). - _Ivan N. Ianakiev_, May 04 2020
%H T. D. Noe, <a href="/A036691/b036691.txt">Table of n, a(n) for n = 0..100</a>
%H Googology Wiki, <a href="http://googology.wikia.com/wiki/Compositorial">Compositorial</a>
%F From _Chayim Lowen_, Jul 23 - Aug 05 2015: (Start)
%F a(n) = A049614(A002808(n)) = A000142(A002808(n))/A034386(A002808(n)).
%F a(n) = Product_{k=1..A002808(n)-n-1} prime(k)^(A085604(A002808(n),k)-1).
%F Sum_{k >= 1} 1/a(k) = 1.2975167655550616507663335821769... is to this sequence as e is to the factorials. (End)
%e a(3) = c(1)*c(2)*c(3) = 4*6*8 = 192.
%p A036691 := proc(n)
%p mul(A002808(i),i=1..n) ;
%p end proc: # _R. J. Mathar_, Oct 03 2011
%t Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[ Product[ Composite[i], {i, 1, n}], {n, 0, 18}] (* _Robert G. Wilson v_, Sep 13 2003 *)
%t nn=50;cnos=Complement[Range[nn],Prime[Range[PrimePi[nn]]]];Rest[FoldList[ Times,1,cnos]] (* _Harvey P. Dale_, May 19 2011 *)
%t A036691 = Union[Table[n!/(Times@@Prime[Range[PrimePi[n]]]), {n, 29}]] (* _Alonso del Arte_, Sep 21 2011 *)
%t Join[{1},FoldList[Times,Select[Range[30],CompositeQ]]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 14 2019 *)
%o (Haskell)
%o a036691_list = scanl1 (*) a002808_list -- _Reinhard Zumkeller_, Oct 03 2011
%o (PARI) a(n)=my(c,p);c=4;p=1;while(n>0,if(!isprime(c),p=p*c;n=n-1);c=c+1);p \\ _Ralf Stephan_, Dec 21 2013
%o (Python)
%o from sympy import factorial, primepi, primorial, composite
%o def A036691(n):
%o return factorial(composite(n))//primorial(primepi(composite(n))) if n > 0 else 1 # _Chai Wah Wu_, Sep 08 2020
%Y Cf. primorial numbers A002110. Distinct members of A049614. See also A049650, A060880.
%Y Cf. A092435 (subsequence: A092435(n) = a(prime(n)-n-1)). - _Chayim Lowen_, Jul 23 2015
%K nice,nonn,easy
%O 0,2
%A _Felice Russo_
%E Corrected and extended by Niklas Eriksen (f95-ner(AT)nada.kth.se) and _N. J. A. Sloane_
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