OFFSET
0,2
COMMENTS
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
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
Googology Wiki, Compositorial
FORMULA
From Chayim Lowen, Jul 23 - Aug 05 2015: (Start)
Sum_{k >= 1} 1/a(k) = 1.2975167655550616507663335821769... is to this sequence as e is to the factorials. (End)
EXAMPLE
a(3) = c(1)*c(2)*c(3) = 4*6*8 = 192.
MAPLE
MATHEMATICA
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 *)
nn=50; cnos=Complement[Range[nn], Prime[Range[PrimePi[nn]]]]; Rest[FoldList[ Times, 1, cnos]] (* Harvey P. Dale, May 19 2011 *)
A036691 = Union[Table[n!/(Times@@Prime[Range[PrimePi[n]]]), {n, 29}]] (* Alonso del Arte, Sep 21 2011 *)
Join[{1}, FoldList[Times, Select[Range[30], CompositeQ]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 14 2019 *)
PROG
(Haskell)
a036691_list = scanl1 (*) a002808_list -- Reinhard Zumkeller, Oct 03 2011
(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
(Python)
from sympy import factorial, primepi, primorial, composite
def A036691(n):
return factorial(composite(n))//primorial(primepi(composite(n))) if n > 0 else 1 # Chai Wah Wu, Sep 08 2020
CROSSREFS
KEYWORD
nice,nonn,easy
AUTHOR
EXTENSIONS
Corrected and extended by Niklas Eriksen (f95-ner(AT)nada.kth.se) and N. J. A. Sloane
STATUS
approved