A049614 -id:A049614 - cp's OEIS frontend

A119490 Sum of the absolute values in row n of A118687.

1, 2, 4, 8, 16, 80, 400, 10000, 250000, 48250000, 83424250000, 1441654464250000, 24913230796704250000, 5166032451235389984250000, 1071233655120621702524064250000, 3109835221395024747917162004384250000, 135419643726614411057926317695276801184250000

Offset: 0

Roger L. Bagula, May 25 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A118687.

A049614[n_]:= n!/Product[Prime[i], {i, 1, PrimePi[n]}];
b:= Join[{{1}}, Table[CoefficientList[Product[1 -A049614[k]*x, {k,0,n}], x], {n, 0, 21}]];
Table[Sum[Abs[b[[n, j]]], {j, n}], {n, 20}] (* G. C. Greubel, Feb 07 2021 *)

def A049614(n): return factorial(n)/product( nth_prime(j) for j in (1..prime_pi(n)) )
def A118687(n,k): return ( product(1 -A049614(k)*x for k in (0..n)) ).series(x, n+2).list()[k]
[1]+[sum(abs(A118687(n, k)) for k in (0..n+1)) for n in (0..20)] # G. C. Greubel, Feb 07 2021

Terms a(12) onward added by G. C. Greubel, Feb 07 2021

A131527 a(n) = 4(n^1 + 1!)(n^2 + 2!)(n^3 + 3!)(n^4 + 4!)*(n^5 + 5!)/5!.

Original entry on oeis.org

1152, 4235, 51072, 1844766, 67267200, 1489787937, 20516082048, 194830108540, 1389727430784, 7923082634775, 37759956198272, 155476758621786, 566979054415488, 1866434208254637, 5629739963760000, 15745829707255032, 41231732634193024, 101887952581305891

Offset: 0

Alexander R. Povolotsky, Aug 25 2007

Comment from Peter J. C. Moses, Aug 29 2007: the values of m = m(k) needed to make the sequence a(n,k) = m (n^1 + 1!) (n^2 + 2!) ... (n^i + k!) / k! (n >= 0) take integral values for all n are given in A049614.

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Table[(Times@@Table[n^k+k!,{k,5}])/30,{n,0,20}] (* Harvey P. Dale, Oct 12 2020 *)
LinearRecurrence[{16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1},{1152,4235,51072,1844766,67267200,1489787937,20516082048,194830108540,1389727430784,7923082634775,37759956198272,155476758621786,566979054415488,1866434208254637,5629739963760000,15745829707255032},30] (* Harvey P. Dale, May 15 2022 *)

G.f.: -(1408*x^14 -221419*x^13 -23074512*x^12 -437328710*x^11 -3130260112*x^10 -9871683909*x^9 -14838023712*x^8 -10832842836*x^7 -3802147872*x^6 -608960101*x^5 -43604624*x^4 -890694*x^3 -121552*x^2 +14197*x -1152) / (x -1)^16. - Colin Barker, Aug 08 2013

A131528 a(n) = (n^1 + 1!)(n^2 + 2!)(n^3 + 3!)*(n^4 + 4!)/2!.

Original entry on oeis.org

144, 525, 5040, 76230, 882000, 6886539, 38974320, 172650300, 633845520, 2008589625, 5657204784, 14470043490, 34161950160, 75378387735, 156979350000, 310979592504, 589757174160, 1076298622245, 1898430030000, 3248190882750, 5407743199824, 8783474489955

Offset: 0

Alexander R. Povolotsky, Aug 25 2007

The values of m = m(k) needed to make the sequence a(n,k) = m (n^1 + 1!) (n^2 + 2!) ... (n^k + k!) / k! (n >= 0) take integral values for all n are given in A049614. - Peter J. C. Moses, Aug 29 2007

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A049614, A131683.

a[n_]:=(n+1)(n^2+2)(n^3+6)(n^4+24)/2;Table[a[n],{n,0,21}] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{144, 525, 5040, 76230, 882000, 6886539, 38974320, 172650300, 633845520, 2008589625,5657204784, 14470043490},22] (* James C. McMahon, Feb 25 2025 *)

G.f.: (-3*(48 + x*(-353 + x*(2395 + x*(8635 + x*(93855 + x*(217437 + x*(213873 + 5*x*(12325 + x*(1441 + 16*x)))))))))) / (x - 1)^11. - Peter J. C. Moses, Aug 29 2007

a(n) = 3*A131683(n). - James C. McMahon, Feb 26 2025

A131682 a(n) = (n^1 + 1!)(n^2 + 2!)(n^3 + 3!)/3!.

Original entry on oeis.org

2, 7, 42, 242, 1050, 3537, 9842, 23732, 51282, 101675, 188122, 328902, 548522, 878997, 1361250, 2046632, 2998562, 4294287, 6026762, 8306650, 11264442, 15052697, 19848402, 25855452, 33307250, 42469427, 53642682, 67165742, 83418442, 102824925, 125856962

Offset: 0

Alexander R. Povolotsky, Aug 29 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

See A049614.

Table[(n^1 + 1!)*(n^2 + 2!)*(n^3 + 3!)/3!, {n,0,50}] (* G. C. Greubel, Feb 21 2017 *)

x='x+O('x^50); Vec((2-7*x+35*x^2+25*x^3+63*x^4+2*x^5)/(1-x)^7) \\ G. C. Greubel, Feb 21 2017

G.f.: (2-7*x+35*x^2+25*x^3+63*x^4+2*x^5)/(1-x)^7. - R. J. Mathar, Nov 14 2007 (Minor edits from Jaume Oliver Lafont, Sep 30 2009)

A131684 a(n) = 24(n^1 + 1!)(n^2 + 2!)(n^3 + 3!)...*(n^6 + 6!)/6!.

Original entry on oeis.org

829440, 3053435, 40040448, 2673065934, 323958835200, 24350583830265, 971969903106048, 23061845117771260, 365309311365605376, 4216355578004498775, 37787143366734755840, 275548505246977513866, 1693398609738955671552, 9010265266941299501973, 42393383140543522560000

Offset: 0

Alexander R. Povolotsky, Aug 29 2007

Colin Barker, Table of n, a(n) for n = 0..1000

See A049614.

a(n) = 24*prod(k=1, 6, n^k + k!)/6!; \\ Michel Marcus, Apr 15 2015

G.f.: -(1103872*x^20 -312478243*x^19 -120671064318*x^18 -10924415464963*x^17 -365289416188928*x^16 -5416497937794108*x^15 -40283003777941144*x^14 -162386613652960172*x^13 -372036568602512352*x^12 -497148917411217834*x^11 -390969147820033860*x^10 -179789868762494298*x^9 -47275965677665472*x^8 -6823180930101292*x^7 -507249401959608*x^6 -17779798800764*x^5 -275765791840*x^4 -1220181963*x^3 -164465518*x^2 +15194245*x -829440) / (x -1)^22. - Colin Barker, Apr 15 2015

A300902 a(n) = n! / Product_{p prime < n}.

Original entry on oeis.org

1, 1, 2, 3, 4, 20, 24, 168, 192, 1728, 17280, 190080, 207360, 2695680, 2903040, 43545600, 696729600, 11844403200, 12541132800, 238281523200, 250822656000, 5267275776000, 115880067072000, 2665241542656000, 2781121609728000, 69528040243200000, 1807729046323200000

Offset: 0

Pedro Caceres, Mar 14 2018

nonn

Sum_{n >= 0} 1/a(n) = 3.1868081118360746...

			a(6) = 6! / Product_{p prime < 6} = 6 * 5 * 4 * 3 * 2/(5 * 3 * 2) = 6 * 4 = 24.

Chai Wah Wu, Table of n, a(n) for n = 0..500

Cf. A000142, A034386, A049614, A066332, A089026.

using Nemo
A300902(n) = div(fac(n), primorial(max(1, n-1)))
[A300902(n) for n in 0:26] |> println # Peter Luschny, Mar 16 2018

a:= n-> n!/mul(`if`(isprime(i), i, 1), i=1..n-1):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 16 2018

Table[n!/(Times@@Prime[Range[PrimePi[n - 1]]]), {n, 0, 29}] (* Alonso del Arte, Mar 25 2018 *)

a(n) = my(v=primes(primepi(n-1))); n!/prod(k=1, #v, v[k]); \\ Michel Marcus, Mar 15 2018

from _future_ import division
from sympy import isprime
A300902_list, m = [1], 1
for n in range(1,501):
    m *= n
    A300902_list.append(m)
    if isprime(n):
        m //= n # Chai Wah Wu, Mar 16 2018

a(n) = A000142(n)/A034386(n-1) for n>0, a(0) = 1.

a(n) = A049614(n)*A089026(n) for n>0, a(0) = 1.

A117733 Sum of the n-th primorial and the n-th compositorial number.

Original entry on oeis.org

2, 3, 7, 10, 34, 54, 234, 402, 1938, 17490, 19590, 209670, 237390, 2933070, 43575630, 696759630, 697240110, 12541643310, 12550832490, 250832355690

Offset: 1

Roger L. Bagula, Apr 14 2006

nonn

The primorial numbers A034386 define their exponential generating function
A034386(x) = sum_{n>=0} A034386(n)*x^n/n! = sum_{n>=0} x^n/A049614(n).
The compositorial numbers A049614 define their exponential generating function
A049614(x) = sum_{n>=0} A049614(n)*x^n/n! = sum_{n>=0} x^n/A034386(n).
Padding the values with A034386(n=0)=A049614(n=0)=1 at the beginning,
two special values of these are
A049614(x=1) = 4.5892461266379861713581024207350707369274... and
A034386(x=1) = 2.9200509773161347120925629171120194680027...

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] a=Table[cf[n] + p[n], {n, 1, 20}]

a(n) = A034386(n)+A049614(n).

Offset and A-number corrected; comment rewritten - The Assoc Eds of the OEIS, Oct 20 2010

A117754 Triangle T(n, k) = (f(n, 1 + (n mod 3)) + f(k, 1 + (k mod 3))) mod n!, read by rows (see formula for f(n, k)).

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 2, 3, 2, 7, 7, 8, 7, 12, 1, 1, 2, 1, 6, 0, 25, 25, 26, 25, 30, 144, 48, 211, 211, 212, 211, 216, 330, 234, 420, 1, 1, 2, 1, 6, 120, 24, 210, 0, 1729, 1729, 1730, 1729, 1734, 1848, 1752, 1938, 42048, 3456, 211, 211, 212, 211, 216, 330, 234, 420, 40530, 1938, 420

Offset: 0

Roger L. Bagula, Apr 14 2006

			Triangle begins as:
     0;
     0,    0;
     1,    1,    0;
     2,    2,    3,    2;
     7,    7,    8,    7,   12;
     1,    1,    2,    1,    6,    0;
    25,   25,   26,   25,   30,  144,   48;
   211,  211,  212,  211,  216,  330,  234,  420;
     1,    1,    2,    1,    6,  120,   24,  210,     0;
  1729, 1729, 1730, 1729, 1734, 1848, 1752, 1938, 42048, 3456;
   211,  211,  212,  211,  216,  330,  234,  420, 40530, 1938, 420;

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

Cf. A034386, A049614, A117682, A117753.

A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
A034386:= func< n | n eq 0 select 1 else LCM(PrimesInInterval(1,n)) >;
function f(n,k)
  if k eq 1 then return A049614(n);
  elif k eq 2 then return A034386(n);
  else return Factorial(n);
  end if;
end function;
A117754:= func< n,k | Floor(f(n, 1+(n mod 3))+f( k, 1+(k mod 3))) mod
 Factorial(n) >;
[A117754(n,k): k in [0..n], n in [0..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 *)
g[n_]:= If[PrimeQ[n], n, 1];
p[n_]:= p[n]= If[n==0, 1, g[n]*p[n-1]];  (* A034386 *)
f[n_, 1]=cf[n]; f[n_, 2]=p[n]; f[n_, 3]=n!;
T[n_, k_]:= Mod[f[n, 1 + Mod[n, 3]] + f[k, 1 + Mod[k, 3]], n!];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

from sympy import primorial
def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1,1+prime_pi(n)))
def A034386(n): return 1 if n == 0 else primorial(n, nth=False)
def f(n,m):
    if m==1: return A049614(n)
    elif m==2: return A034386(n)
    else: return factorial(n)
def A117754(n, k): return (f(n, 1+(n%3))+f(k, 1+(k%3)))%factorial(n)
flatten([[A117754(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 21 2023

T(n, k) = (f(n, 1 + (n mod 3)) + f(k, 1 + (k mod 3))) mod n!, where f(n, 1) = A049614(n), f(n, 2) = A034386(n), and f(n, 3) = n!.

Edited by G. C. Greubel, Jul 21 2023

A271387 Numerator of prime(n)#/n!, where prime(n)# is the prime factorial function.

Original entry on oeis.org

1, 2, 3, 5, 35, 77, 1001, 2431, 46189, 1062347, 30808063, 86822723, 3212440751, 10131543907, 435656388001, 20475850236047, 1085220062510491, 3766351981654057, 229747470880897477, 810162134158954261, 57521511525285752531, 4199070341345859934763, 331726556966322934846277

Offset: 0

Ilya Gutkovskiy, Apr 06 2016

			1, 2, 3, 5, 35/4, 77/4, 1001/24, 2431/24, 46189/192, 1062347/1728, 30808063/17280, 86822723/17280, 3212440751/207360, 10131543907/207360, 435656388001/2903040, ...
a(8) = 46189, because prime(8)#/8! = (2*3*5*7*11*13*17*19)/(1*2*3*4*5*6*7*8) = 46189/192.

Ilya Gutkovskiy, Table of n, a(n) for n = 0..75
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Factorial

Cf. A000040, A000142, A000720, A002110, A007947, A034386, A049614 (denominator of prime(n)#/n!), A090586, A135568.

Table[Numerator[Product[Prime@ k, {k, n}]/n!], {n, 0, 22}] (* Michael De Vlieger, Apr 08 2016 *)

a(n) = numerator(prod(k=1, n, prime(k))/n!); \\ Michel Marcus, Apr 09 2016

a(n) = prime(n)#/GCD(prime(n)#, n!), where GCD(a, b) is the greatest common divisor.
a(n) = prime(n)#/prime(pi(n))#, where pi(n) is the number of primes <= n.
a(n) = A002110(n)/A034386(n) = A002110(n)/A002110(A000720(n)) = A002110(n)/A007947(A000142(n)).

A085274 Composite k such that (k!/k#) + 1 is a semiprime, where k# = primorial numbers A034386.

Original entry on oeis.org

6, 10, 21, 22, 24, 25, 27, 30, 39, 48, 52, 57, 65, 87, 94, 110, 114, 124, 156, 161

Offset: 1

Jason Earls, Aug 12 2003

nonn

n!/n# is called n compositorial. The actual sequence is (6,7),(10,11),21,(22,23),24,25,27,(30,31),39,48, (52,53),57,65,... where the values in parenthesis yield the same semiprime.

That is, since p!/p# = (p-1)!/(p-1)#, primes never appear in this sequence. - Sean A. Irvine, Jun 30 2020

			25!/25# + 1 is a product of two primes: 69528040243200001 = 2594807 * 26795071943.

Hisanori Mishima, Compositorial + 1 (n = 4 to 150).

Cf. A001358, A049420, A049614.

More terms from Robert G. Wilson v, Aug 15 2003

Offset corrected and a(19) and a(20) from Sean A. Irvine, Jun 30 2020

cp's OEIS Frontend

A119490 Sum of the absolute values in row n of A118687.

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A131527 a(n) = 4*(n^1 + 1!)*(n^2 + 2!)*(n^3 + 3!)*(n^4 + 4!)*(n^5 + 5!)/5!.

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A131528 a(n) = (n^1 + 1!)*(n^2 + 2!)*(n^3 + 3!)*(n^4 + 4!)/2!.

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A131682 a(n) = (n^1 + 1!)*(n^2 + 2!)*(n^3 + 3!)/3!.

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A131684 a(n) = 24*(n^1 + 1!)*(n^2 + 2!)*(n^3 + 3!)*...*(n^6 + 6!)/6!.

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A300902 a(n) = n! / Product_{p prime < n}.

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A117733 Sum of the n-th primorial and the n-th compositorial number.

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A117754 Triangle T(n, k) = (f(n, 1 + (n mod 3)) + f(k, 1 + (k mod 3))) mod n!, read by rows (see formula for f(n, k)).

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A131527 a(n) = 4(n^1 + 1!)(n^2 + 2!)(n^3 + 3!)(n^4 + 4!)*(n^5 + 5!)/5!.

A131528 a(n) = (n^1 + 1!)(n^2 + 2!)(n^3 + 3!)*(n^4 + 4!)/2!.

A131682 a(n) = (n^1 + 1!)(n^2 + 2!)(n^3 + 3!)/3!.

A131684 a(n) = 24(n^1 + 1!)(n^2 + 2!)(n^3 + 3!)...*(n^6 + 6!)/6!.