A049614 -id:A049614 - cp's OEIS frontend

A117683 Triangle T(n,k) = A049614(n)/(A049614(k)*A049614(n-k)), read by rows.

1, 1, 1, 1, 1, 1, 4, 4, 4, 1, 1, 4, 4, 1, 1, 6, 6, 24, 6, 6, 1, 1, 6, 6, 6, 6, 1, 1, 8, 8, 48, 12, 48, 8, 8, 1, 9, 72, 72, 108, 108, 72, 72, 9, 1, 10, 90, 720, 180, 1080, 180, 720, 90, 10, 1, 1, 10, 90, 180, 180, 180, 180, 90, 10, 1, 1, 12, 12, 120, 270, 2160, 360, 2160, 270, 120, 12, 12, 1

Offset: 1

Roger L. Bagula, Apr 12 2006

			Triangle begins as:
  1;
  1,  1;
  1,  1,  1;
  4,  4,  4,   1;
  1,  4,  4,   1,   1;
  6,  6, 24,   6,   6,  1;
  1,  6,  6,   6,   6,  1,  1;
  8,  8, 48,  12,  48,  8,  8,  1;
  9, 72, 72, 108, 108, 72, 72,  9,  1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A049614, A117684.

A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
A117683:= func< n,k | A049614(n)/(A049614(k)*A049614(n-k)) >;
[A117683(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 21 2023

f[n_]:= If[PrimeQ[n], 1, n];
cf[n_]:= cf[n]= If[n==0, 1, f[n]*cf[n-1]]; (* A049614 *)
T[n_, k_]:= T[n, k]= cf[n]/(cf[k]*cf[n-k]);
Table[T[n, k], {n,12}, {k,n}]//Flatten

primorial(n)=prod(i=1,primepi(n),prime(i))
T(n,m)=binomial(n,m)*primorial(m)*primorial(n-m)/primorial(n) \\ Charles R Greathouse IV, Jan 16 2012

def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1, 1+prime_pi(n)))
def A117683(n,k): return A049614(n)/(A049614(k)*A049614(n-k))
flatten([[A117683(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jul 21 2023

T(n,k) = A049614(n)/(A049614(k)*A049614(n-k)), for 1 <= k <= n, n >= 1.

Sum_{k=1..n} T(n, k) = A117684(n).

Edited by the Associate Editors of the OEIS, Aug 18 2009

Edited by G. C. Greubel, Jul 21 2023

A118687 A triangular array made from polynomial coefficients of A049614.

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -3, 3, -1, 1, -4, 6, -4, 1, 1, -8, 22, -28, 17, -4, 1, -12, 54, -116, 129, -72, 16, 1, -36, 342, -1412, 2913, -3168, 1744, -384, 1, -60, 1206, -9620, 36801, -73080, 77776, -42240, 9216, 1, -252, 12726, -241172, 1883841, -7138872, 14109136, -14975232, 8119296, -1769472

Offset: 0

Roger L. Bagula, May 20 2006

Same as an alternating sign Pascal's triangle up to row n=4.

			Triangle begins as:
  1;
  1, -1;
  1, -2,  1;
  1, -3,  3, -1;
  1, -4,  6, -4,  1;
  1, -8, 22,-28, 17, -4;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A008275, A034386, A049614, A119490.

A049614[n_]:= n!/Product[Prime[i], {i, 1, PrimePi[n]}];
Join[{{1}}, Table[CoefficientList[Product[1 - A049614[k]*x, {k, 0, n}], x], {n, 0, 12}]]//Flatten

def A049614(n): return factorial(n)/product( nth_prime(j) for j in (1..prime_pi(n)) )
[1]+flatten([[( product(1 - A049614(k)*x for k in (0..n)) ).series(x,n+2).list()[k] for k in (0..n+1)] for n in (0..12)]) # G. C. Greubel, Feb 05 2021

T(n, k) = coefficients of Product_{k=0..n} (1 - A049614(k)*x), with T(0, 0) = 1.

Edited by G. C. Greubel, Feb 05 2021

A078744 Numbers k such that A049614(k) + A000040(k) is prime.

Original entry on oeis.org

1, 4, 6, 7, 8, 13, 17, 18, 21, 41, 86, 101, 144, 368, 1232, 1713, 2431, 2897, 6122, 6240, 6876

Offset: 1

Jason Earls, Dec 21 2002

Cf. A000040, A034386, A049614, A078745.

is(k) = ispseudoprime(prod(i=1, k, i^!isprime(i)) + prime(k)); \\ Jinyuan Wang, Jan 08 2021

a(17)-a(18) from Jinyuan Wang, Jan 08 2021

a(19)-a(21) from Michael S. Branicky, Aug 10 2024

A078745 Numbers k such that A049614(k) - A000040(k) is prime.

Original entry on oeis.org

6, 7, 8, 15, 23, 36, 93, 117, 121, 131, 150, 310, 433, 896, 1226, 4439, 11452, 13069

Offset: 1

Jason Earls, Dec 21 2002

Cf. A000040, A034386, A049614, A078744.

is(k) = ispseudoprime(prod(i=1, k, i^!isprime(i)) - prime(k)); \\ Jinyuan Wang, Jan 09 2021

a(16) from Ryan Propper, Feb 12 2007

a(17)-a(18) from Michael S. Branicky, Aug 10 2024

A082956 Difference between C(n) and the first prime after C(n), where C(n) = A049614(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 5, 1, 5, 11, 11, 7, 7, 1, 11, 29, 29, 17, 17, 1, 47, 17, 17, 37, 79, 1, 31, 397, 397, 47, 47, 53, 127, 1, 53, 67, 67, 41, 23, 23, 23, 41, 41, 113, 107, 31, 31, 79, 157, 61, 353, 157, 157, 601, 31, 1, 29, 409, 409, 31, 31, 181, 41, 139, 47, 151, 151, 47, 47, 109

Offset: 1

Jason Earls, May 26 2003

nonn

All terms here are either 1 or prime.

Bill McEachen, Table of n, a(n) for n = 1..1000

Cf. A049614.

A082956[n_] := NextPrime[#] - # & [n!/Product[Prime[i], {i, PrimePi[n]}]];
Array[A082956, 100] (* Paolo Xausa, Aug 02 2024 *)

a(n)= my(C=n!/prod(i=1, primepi(n), prime(i))); nextprime(C) - C; \\ Michel Marcus, Jul 31 2024

A117878 Triangle T(n,k) = A034386(n)*A049614(k) - 1 read by rows.

Original entry on oeis.org

0, 1, 1, 5, 5, 5, 5, 5, 5, 23, 29, 29, 29, 119, 119, 29, 29, 29, 119, 119, 719, 209, 209, 209, 839, 839, 5039, 5039, 209, 209, 209, 839, 839, 5039, 5039, 40319, 209, 209, 209, 839, 839, 5039, 5039, 40319, 362879, 209, 209, 209, 839, 839, 5039, 5039, 40319, 362879

Offset: 1

Roger L. Bagula, May 02 2006

			The triangle starts in row n=1 as:
    0;
    1,   1;
    5,   5,   5;
    5,   5,   5,  23;
   29,  29,  29, 119, 119;
   29,  29,  29, 119, 119,  719;
  209, 209, 209, 839, 839, 5039, 5039;
  209, 209, 209, 839, 839, 5039, 5039, 40319;
  209, 209, 209, 839, 839, 5039, 5039, 40319, 362879;
  209, 209, 209, 839, 839, 5039, 5039, 40319, 362879, 3628799;

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A034386, A049614.

A034386[n_]:= Product[Prime[i], {i, PrimePi[n]}];
A049614[n_]:= n!/A034386[n];
Table[A034386[n]*A049614[k] - 1, {n, 10}, {k, n}]//Flatten (* G. C. Greubel, Feb 06 2021 *)

def A034386(n): return product( nth_prime(j) for j in (1..prime_pi(n)) )
def A117878(n, k): return factorial(k)*A034386(n)/A034386(k) - 1
flatten([[A117878(n,k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Feb 06 2021

T(n, k) = A034386(n)*A049614(k) - 1.

T(n, k) = k! * A034386(n)/A034386(k) - 1 = n! * A049614(k)/A049614(n) - 1. - G. C. Greubel, Feb 06 2021

Index in definition and offset corrected by Assoc. Eds. of the OEIS - Jun 15 2010

A117736 factorial(n) - A049614(n).

Original entry on oeis.org

0, 0, 1, 5, 20, 116, 696, 5016, 40128, 361152, 3611520, 39899520, 478794240, 6226813440, 87175388160, 1307630822400, 20922093158400, 355686731366400, 6402361164595200, 121645087867699200, 2432901757353984000

Offset: 0

Roger L. Bagula, Apr 14 2006

nonn

Cf. A034386, A117683.

f[n_] := If[PrimeQ[n] == True, 1, n] cf[0] = 1; cf[n_Integer?Positive] := cf[n] = f[n]*cf[n - 1] g[n_] := If[PrimeQ[n] == True, n, 1] p[0] = 1; p[n_Integer?Positive] := p[n] = g[n]*p[n - 1] Table[n! - cf[n], {n, 0, 20}]

a(n) =n!- A049614(n) = A000142(n)-A049614(n).

Edited by the Assoc. Eds. of the OEIS, Jun 27 2010

A131978 A049614(n)/A131685(n).

Original entry on oeis.org

1, 1, 1, 4, 4, 24, 24, 192, 1728, 17280, 17280, 207360, 207360, 414720, 6220800, 99532800, 696729600, 12541132800, 12541132800, 250822656000, 5267275776000, 10534551552000, 10534551552000, 252829237248000, 1264146186240000, 12641461862400000, 3754514173132800000

Offset: 1

Alexander R. Povolotsky, Sep 21 2007

nonn

Conjectured to be always integral.

It appears that every term > 4 is divisible by 3 - Alexander R. Povolotsky, Oct 18 2007

A172985 Triangle T(n,m) = A049614(n)/ (A049614(m)*A049614(n-m)) read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 1, 1, 1, 4, 4, 1, 1, 1, 6, 6, 24, 6, 6, 1, 1, 1, 6, 6, 6, 6, 1, 1, 1, 8, 8, 48, 12, 48, 8, 8, 1, 1, 9, 72, 72, 108, 108, 72, 72, 9, 1, 1, 10, 90, 720, 180, 1080, 180, 720, 90, 10, 1, 1, 1, 10, 90, 180, 180, 180, 180, 90, 10, 1, 1, 1, 12, 12, 120, 270

Offset: 0

Roger L. Bagula, Feb 06 2010

Row sums are: 1, 2, 3, 4, 14, 12, 50, 28, 142, 524, 3082,...

			1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 4, 4, 4, 1;
1, 1, 4, 4, 1, 1;
1, 6, 6, 24, 6, 6, 1;
1, 1, 6, 6, 6, 6, 1, 1;
1, 8, 8, 48, 12, 48, 8, 8, 1;
1, 9, 72, 72, 108, 108, 72, 72, 9, 1;
1, 10, 90, 720, 180, 1080, 180, 720, 90, 10, 1;

c[n_] := Product[If[i == 0, 1, If[PrimeQ[i], 1, i]], {i, 0, n}];
t[n_, m_] := c[n]/(c[m]*c[n - m]);
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Definition simplified by the Assoc. Editors of the OEIS, Mar 05 2010

A036691 Compositorial numbers: product of first n composite numbers.

Original entry on oeis.org

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

cp's OEIS Frontend

A117683 Triangle T(n,k) = A049614(n)/(A049614(k)*A049614(n-k)), read by rows.

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A118687 A triangular array made from polynomial coefficients of A049614.

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A078744 Numbers k such that A049614(k) + A000040(k) is prime.

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A078745 Numbers k such that A049614(k) - A000040(k) is prime.

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A082956 Difference between C(n) and the first prime after C(n), where C(n) = A049614(n).

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A117878 Triangle T(n,k) = A034386(n)*A049614(k) - 1 read by rows.

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A117736 factorial(n) - A049614(n).

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A131978 A049614(n)/A131685(n).

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