A049650 -id:A049650 - cp's OEIS frontend

A036691 Compositorial numbers: product of first n composite numbers.

1, 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, 12541132800, 250822656000, 5267275776000, 115880067072000, 2781121609728000, 69528040243200000, 1807729046323200000, 48808684250726400000, 1366643159020339200000

Offset: 0

Felice Russo

a(A196415(n)) = A141092(n) * A053767(A196415(n)). - Reinhard Zumkeller, Oct 03 2011
For n>11, A000142(n) < a(n) < A002110(n). - Chayim Lowen, Aug 18 2015
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

			a(3) = c(1)*c(2)*c(3) = 4*6*8 = 192.

T. D. Noe, Table of n, a(n) for n = 0..100
Googology Wiki, Compositorial

Cf. primorial numbers A002110. Distinct members of A049614. See also A049650, A060880.

Cf. A092435 (subsequence: A092435(n) = a(prime(n)-n-1)). - Chayim Lowen, Jul 23 2015

a036691_list = scanl1 (*) a002808_list -- Reinhard Zumkeller, Oct 03 2011

A036691 := proc(n)
        mul(A002808(i),i=1..n) ;
end proc: # R. J. Mathar, Oct 03 2011

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 *)

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

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

From Chayim Lowen, Jul 23 - Aug 05 2015: (Start)
a(n) = A049614(A002808(n)) = A000142(A002808(n))/A034386(A002808(n)).
a(n) = Product_{k=1..A002808(n)-n-1} prime(k)^(A085604(A002808(n),k)-1).
Sum_{k >= 1} 1/a(k) = 1.2975167655550616507663335821769... is to this sequence as e is to the factorials. (End)

Corrected and extended by Niklas Eriksen (f95-ner(AT)nada.kth.se) and N. J. A. Sloane

A053982 Numbers k such that 1 + product of first k composite numbers is prime.

Original entry on oeis.org

1, 3, 7, 11, 16, 22, 39, 76, 116, 139, 149, 169, 179, 220, 372, 429, 1216, 2146, 3176, 5382, 5969, 12271, 15271, 43903

Offset: 1

G. L. Honaker, Jr., Apr 02 2000

Cf. A002808, A000040, A049420, A049650, A057017, A140294.

Composite[n_Integer] := (k = n + PrimePi[n] + 1; While[k - PrimePi[k] - 1 != n, k++ ]; k); Do[ If[ PrimeQ[ Product[ Composite[k], {k, 1, n} ] + 1], Print[ n ] ], {n, 1, 430} ]
Position[FoldList[Times,Select[Range[1500],CompositeQ]],?(PrimeQ[#+1]&)]//Flatten (* _Harvey P. Dale, Dec 20 2022 *)

lista(kmax) = {my(m = 1, k = 0); forcomposite(c = 1, , k++; if(k > kmax, break); m *= c; if(isprime(m+1), print1(k, ", ")));} \\ Amiram Eldar, Jun 03 2024

More terms from Jeppe Stig Nielsen, Apr 16 2000 (terms from 76 on correspond to probable primes)
a(16)-a(17) from Robert G. Wilson v, Apr 20 2001
Edited by T. D. Noe, Oct 30 2008
a(18)-a(19) from Amiram Eldar, Jun 03 2024
a(20)-a(21) from Michael S. Branicky, Jun 04 2024
More terms via A049420 from Jeppe Stig Nielsen, Aug 12 2024

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A036691 Compositorial numbers: product of first n composite numbers.

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A053982 Numbers k such that 1 + product of first k composite numbers is prime.

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A060880 Compositorial numbers (A036691) - 1.

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