A049614 -id:A049614 - cp's OEIS frontend

A131685 a(n) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^n + n) / n! takes integral values for all i>=0.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 1, 1, 1, 1, 1, 11, 11, 11, 55, 143, 13, 91, 91, 91, 91, 91, 1001, 17017, 595595, 595595, 17017, 46189, 600457, 3002285, 3002285, 3002285, 3002285, 6605027, 3002285, 726869, 726869, 726869

Offset: 1

Alexander R. Povolotsky and Peter J. C. Moses, Sep 12 2007, revised Sep 17 2007

nonn

It appears that none of the terms are divisible by 3. - Alexander R. Povolotsky, Oct 18 2007

Cyril Banderier, Table of n, a(n) for n = 1..100
Index to divisibility sequences

Cf. A000027 (for n=1), A064808 (n=2), A131509 (n=3), A129995 (n=4), A131675 (n=5), ..., A131680 (n=10).

A135291 Product of the nonzero exponents in the prime factorization of n!.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 8, 8, 14, 28, 64, 64, 100, 100, 220, 396, 540, 540, 768, 768, 1152, 1944, 4104, 4104, 5280, 7920, 16560, 21528, 31200, 31200, 40768, 40768, 48608, 78120, 161280, 230400, 277440, 277440, 571200, 907200, 1108080, 1108080, 1440504, 1440504, 2019168

Offset: 0

Leroy Quet, Dec 03 2007

nonn

a(n) = A005361(n!). For n >= 2, a(n) = the number of positive divisors of n! which themselves are each divisible by every prime <= n. For p = any prime, a(p) = a(p-1). a(0)=a(1)=1 because the product of the exponents is over the empty set.

			6! = 720 has a prime factorization of 2^4 * 3^2 * 5^1. So a(6) = 4*2*1 = 8.
Also, 720 is divisible by a(6)=8 positive divisors which themselves are each divisible by every prime <= 6 (i.e., are each divisible by 2*3*5 = 30): 30, 60, 90, 120, 180, 240, 360, 720.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from G. C. Greubel)
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Cf. A005361, A000005, A049614, A085361.

A005361 := proc(n) mul( op(2,i),i=ifactors(n)[2]) ; end: A135291 := proc(n) A005361(n!) ; end: seq(A135291(n),n=0..50) ; # R. J. Mathar, Dec 12 2007
# second Maple program:
b:= proc(n) option remember; `if`(n<1, 1,
      b(n-1)+add(i[2]*x^i[1], i=ifactors(n)[2]))
    end:
a:= n-> mul(i, i=coeffs(b(n))):
seq(a(n), n=0..44);  # Alois P. Heinz, Jun 02 2025

Table[Product[FactorInteger[n! ][[i, 2]], {i, 1, Length[FactorInteger[n! ]]}], {n, 0, 50}] (* Stefan Steinerberger, Dec 05 2007 *)
Table[Times@@Transpose[FactorInteger[n!]][[2]],{n,0,50}] (* Harvey P. Dale, Aug 16 2011 *)

valp(n,p)=my(s); while(n\=p, s+=n); s
a(n)=my(s=1); forprime(p=2,n\2, s*=valp(n,p)); s \\ Charles R Greathouse IV, Oct 09 2016

from math import prod
from collections import Counter
from sympy import factorint
def A135291(n): return prod(sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values()) # Chai Wah Wu, Jun 02 2025

a(n) = A000005(A049614(n)). - Ridouane Oudra, Sep 02 2019

a(n) = exp((n/log(n)) * (Sum_{k=0..M} e_k/log(n)^k) + O(n/log(n)^(M+2))) for any given integer M >= 0, where e_k = k! * Sum_{j=0..k} (1/j!) * Sum_{s>=1} (log(s+1)^j/(s+1))*log(1+1/s) are constants (e_0 = A085361) (De Koninck and Verreault, 2024, p. 54, Theorem 4.4). - Amiram Eldar, Dec 10 2024

More terms from Stefan Steinerberger and R. J. Mathar, Dec 05 2007

A092435 Prime factorials divided by their corresponding primorials.

Original entry on oeis.org

1, 1, 4, 24, 17280, 207360, 696729600, 12541132800, 115880067072000, 1366643159020339200000, 40999294770610176000000, 1854768736099424576471040000000, 109950690675973888893203251200000000, 4617929008390903333514536550400000000, 420600974084243475616503989010432000000000

Offset: 1

Don Willard (dwillard(AT)prairie.cc.il.us), Mar 23 2004

nonn

			E.g., 2 factorial divided by 2 primorial is 1; 3 factorial is 6, divided by 3 primorial (3*2=6) is also 1; 5 factorial is 120, divided by 5 primorial (5*3*2=30) is 4 and so forth.

Subsequence of A036691. - Chayim Lowen, Jul 23 2015

Cf. A002110, A039716.

a:= proc(n) option remember; `if`(n<2, 1,
      a(n-1)*mul(i, i=ithprime(n-1)+1..ithprime(n)-1))
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Jan 15 2025

Table[ Prime[n]! / Times @@ Prime[ Range[ n]], {n, 13}] (* Robert G. Wilson v, Mar 25 2004 *)

a(n)=prime(n)!/prod(i=1,n,prime(i)) \\ Ralf Stephan, Dec 21 2013

p!/p# = A039716/A002110.
Partial products of A061214. - Lekraj Beedassy, Nov 06 2006
From Chayim Lowen, Jul 23 - Aug 05 2015: (Start)
a(n) = A036691(A065890(n)).
a(n) = A000142(A002808(A065890(n)))/A034386(A002808(A065890(n))).
a(n) = Product_{k=1..n} prime(k)^(A085604(prime(n),k)-1).
a(n) = A049614(prime(n)).
a(n) = Product_{k=1..prime(n)} k^A066247(k). (End)

Edited by Robert G. Wilson v, Mar 25 2004

More terms from Michel Marcus, Jan 15 2025

A140293 Numbers k such that k!/k#-1 is prime, where k# is the primorial function (A034386).

Original entry on oeis.org

4, 5, 6, 7, 8, 16, 17, 21, 34, 39, 45, 50, 72, 73, 76, 133, 164, 202, 216, 221, 280, 281, 496, 605, 2532, 2967, 3337, 8711, 10977, 13724, 15250, 18160, 20943, 33684, 41400

Offset: 1

Cino Hilliard, May 25 2008

a(31) > 14000. - Giovanni Resta, Apr 02 2013
a(36) > 50000. - Roger Karpin, Jul 07 2015
If k is a prime and k is a member, then k-1 is also a member, and k!/k# - 1 is the same as (k-1)!/(k-1)# - 1. See A049421. - Jeppe Stig Nielsen, Aug 12 2024

			7!/7# = 5040/210 = 24. 24 - 1 = 23, which is prime.

Chris Caldwell, Compositorial
Chris Caldwell, Compositorial list search

Cf. A140294, A140315, A049420, A049421, A049614, A057017.

Select[Range[16], PrimeQ[#!/(Times@@Prime[Range[PrimePi[#]]]) - 1] &] (* Alonso del Arte, Nov 28 2014 *)

g(n) = for(x=4,n,y=x!/primorial(x)-1;z=nextprime(y+1); if(ispseudoprime(y),print1(x",")))

n such that n!/n# - 1 is prime, where n# is the primorial function n# = product(i = 1 .. pi(n), prime(i)), where pi(n) is the prime counting function.

a(18)-a(27) from Giovanni Resta, Mar 28 2013
a(28)-a(30) from Giovanni Resta, Apr 02 2013
a(31) from Roger Karpin, Nov 28 2014
a(32)-a(33) from Daniel Heuer, ca Aug 2000
a(34)-a(35) from Serge Batalov, Feb 09 2015

A140315 Numbers k such that k!/k#-1 and k!/k#+1 are a twin prime pair.

Original entry on oeis.org

4, 5, 8, 34, 280, 281

Offset: 1

Cino Hilliard, May 25 2008

4,5 and 280,281 result in the same respective twin prime pairs. Using gmp, testing n < 4000, the last 3-prp found was the 8897 digit 3-prp, 3337!/3337#-1.

			8!/8#-1 = 191, 8!/8#-1 = 193. 191 and 193 form a twin prime pair.

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 281.

Cf. A001097, A034386, A049614, A140293, A140294.

Primorial[n_] := Product[Prime[i], {i, 1, PrimePi[n]}];
Select[Range[
  1000], (p = (#! / Primorial[#]);
PrimeQ[p + 1] && PrimeQ[p - 1]) &] (* Robert Price, Oct 11 2019 *)

g(n) = for(x=1,n,y=x!/primorial(x)-1;z=nextprime(y+1); if(ispseudoprime(y)&&z-y==2,print1(x", ")))
primorial(n) = { local(p1,x); if(n==0||n==1,return(1)); p1=1; forprime(x=2,n,p1*=x); return(p1) }

n# is the primorial function A034386(n).

A140293 INTERSECT A140294. - R. J. Mathar, Feb 27 2012

Offset corrected by Amiram Eldar, Jul 18 2025

A131683 a(n) = 4(n^1 + 1!)(n^2 + 2!)(n^3 + 3!)(n^4 + 4!)/4!.

Original entry on oeis.org

48, 175, 1680, 25410, 294000, 2295513, 12991440, 57550100, 211281840, 669529875, 1885734928, 4823347830, 11387316720, 25126129245, 52326450000, 103659864168, 196585724720, 358766207415, 632810010000, 1082730294250, 1802581066608, 2927824829985, 4650083620560

Offset: 0

Alexander R. Povolotsky, 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).

See A049614.

Table[((n+1)(n^2+2)(n^3+6)(n^4+24))/6,{n,0,30}] (* or *) LinearRecurrence[ {11,-55,165,-330,462,-462,330,-165,55,-11,1},{48,175,1680,25410,294000,2295513,12991440,57550100,211281840,669529875,1885734928},30] (* Harvey P. Dale, Mar 23 2015 *)

G.f.: -(80*x^9 +7205*x^8 +61625*x^7 +213873*x^6 +217437*x^5 +93855*x^4 +8635*x^3 +2395*x^2 -353*x +48) / (x -1)^11. - Colin Barker, Aug 08 2013

a(0)=48, a(1)=175, a(2)=1680, a(3)=25410, a(4)=294000, a(5)=2295513, a(6)=12991440, a(7)=57550100, a(8)=211281840, a(9)=669529875, a(10)=1885734928, a(n)=11*a(n-1)- 55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)- 165*a(n-8)+ 55*a(n-9)- 11*a(n-10)+a(n-11). - Harvey P. Dale, Mar 23 2015

A066332 a(1)=1; for n > 0, a(n+1) = rad(a(n))*n where rad=A007947.

Original entry on oeis.org

1, 1, 2, 6, 24, 30, 180, 210, 1680, 1890, 2100, 2310, 27720, 30030, 420420, 450450, 480480, 510510, 9189180, 9699690, 193993800, 203693490, 213393180, 223092870, 5354228880, 5577321750, 5800414620, 6023507490, 6246600360, 6469693230, 194090796900, 200560490130

Offset: 1

Reinhard Zumkeller, Jan 05 2002

nonn

Cf. A002110, A008864, A049614.

rad[n_] := Times @@ (First@# & /@ FactorInteger@n); a[n_] := (n -1)rad[a[n -1]]; a[1] = 1; Array[a, 30] (* Robert G. Wilson v, Dec 14 2016 and modified Dec 24 2016 *)

a(n) = if(n==1, 1, (n-1)*prod(k=1, primepi(n-2), prime(k))); \\ Daniel Suteu, Dec 14 2016

rad(n) = factorback(factorint(n)[, 1]);
a(n) = if (n==1, 1, (n-1)*rad(a(n-1))); \\ Michel Marcus, Dec 21 2016

a(1)=1; for n > 1, a(n) = (n-1) * Product_{p prime < (n-1)} p. - Pedro Caceres, Mar 12 2018

a(A008864(n)) = A002110(n). - Michel Marcus, Mar 17 2018

More terms from Michel Marcus, Mar 17 2018

A073840 Product of the composite numbers between n and 2n (both inclusive).

Original entry on oeis.org

1, 4, 24, 192, 4320, 51840, 120960, 29030400, 65318400, 145152000, 6706022400, 160944537600, 8717829120000, 6590678814720000, 14122883174400000, 30128817438720000, 2112783322890240000, 2662106986841702400000

Offset: 1

Amarnath Murthy, Aug 13 2002

nonn

a(n) is divisible by central binomial coefficients, A001405[n]

			a(6) = 6*8*9*10*12 = 51840.

Cf. A073839, A073641.

for n from 1 to 50 do l := 1:for j from n to 2*n do if not isprime(j) then l := l*j:fi:od:a[n] := l:od:seq(a[j],j=1..50);

cs[x_] := Flatten[Position[Table[PrimeQ[j], {j, x, 2*x}], False]]+x-1; prcs[x_] := Apply[Times, cs[x]]; Table[prcs[w], {w, 1, 25}]

PARI
```
a(n)=prod(i=n,2*n,i^if(isprime(i),0,1))
```

a(n)=A049614(2n)/A049614(n-1)

More terms from Sascha Kurz and Labos Elemer, Aug 14 2002

A117684 Row sums of A117683.

Original entry on oeis.org

1, 2, 3, 13, 11, 49, 27, 141, 523, 3081, 923, 5509, 1371, 7617, 24391, 84933, 14795, 110329, 20859, 142101, 499843, 1858209, 241211, 2312077, 8417451, 70482153, 251680159, 935093181, 95916299, 1102272481, 131510523, 1270525629, 4572551611, 17189356473

Offset: 1

Roger L. Bagula, Apr 12 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A117683.

A034386:= func< n | n eq 0 select 1 else LCM(PrimesInInterval(1, n)) >;
[(&+[Binomial(n,k)*A034386(k)*A034386(n-k)/A034386(n): k in [1..n]]): n in [1..40]]; // 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]);
a[n_]:= a[n]= Sum[T[n,k], {k,n}];
Table[a[n], {n,40}]

@CachedFunction
def A034386(n): return product(nth_prime(j) for j in range(1, 1+prime_pi(n)))
def A117684(n): return sum(binomial(n,k)*A034386(k)*A034386(n-k)/A034386(n) for k in range(1,n+1))
[A117684(n) for n in range(1,41)] # G. C. Greubel, Jul 21 2023

a(n) = Sum_{k=1..n} A117683(n,k).

Description simplified, offset corrected by the Assoc. Eds. of the OEIS, Jun 27 2010

A117753 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, 0, 0, 0, 1, 1, 2, 1, 6, 6, 12, 6, 12, 0, 0, 0, 0, 0, 0, 24, 24, 48, 24, 144, 0, 576, 210, 210, 420, 210, 1260, 0, 0, 3780, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1728, 1728, 3456, 1728, 10368, 207360, 41472, 0, 0, 82944, 210, 210, 420, 210, 1260, 25200, 5040, 44100, 1209600, 362880, 44100

Offset: 0

Roger L. Bagula, Apr 14 2006

			Triangle begins as:
     0;
     0,    0;
     0,    0,    0;
     1,    1,    2,    1;
     6,    6,   12,    6,    12;
     0,    0,    0,    0,     0,      0;
    24,   24,   48,   24,   144,      0,   576;
   210,  210,  420,  210,  1260,      0,     0,  3780;
     0,    0,    0,    0,     0,      0,     0,     0,   0;
  1728, 1728, 3456, 1728, 10368, 207360, 41472,     0,   0, 82944;

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

Cf. A034386, A049614, A117682.

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;
A117753:= func< n,k | Floor( f( n, 1 + (n mod 3) )*f( k, 1 + (k mod 3)) ) mod Factorial(n) >;
[A117753(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!;
Table[Mod[f[n, 1 + Mod[n, 3]]*f[m, 1 + Mod[m, 3]], n!], {n, 0, 10}, {m, 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 A117753(n, k): return (f(n, 1+(n%3))*f(k, 1+(k%3)))%factorial(n)
flatten([[A117753(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

cp's OEIS Frontend

A131685 a(n) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^n + n) / n! takes integral values for all i>=0.

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A135291 Product of the nonzero exponents in the prime factorization of n!.

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A092435 Prime factorials divided by their corresponding primorials.

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A140293 Numbers k such that k!/k#-1 is prime, where k# is the primorial function (A034386).

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A140315 Numbers k such that k!/k#-1 and k!/k#+1 are a twin prime pair.

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A131683 a(n) = 4*(n^1 + 1!)*(n^2 + 2!)*(n^3 + 3!)*(n^4 + 4!)/4!.

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A066332 a(1)=1; for n > 0, a(n+1) = rad(a(n))*n where rad=A007947.

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A131683 a(n) = 4(n^1 + 1!)(n^2 + 2!)(n^3 + 3!)(n^4 + 4!)/4!.