A088140 -id:A088140 - cp's OEIS frontend

A005451 a(n) = 1 if n is a prime number, otherwise a(n) = n.

1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60

Offset: 1

N. J. A. Sloane

Denominator of (1 + Gamma(n))/n.

Möbius transform of A380441(n). - Wesley Ivan Hurt, Jun 21 2025

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Achilleas Sinefakopoulos, Problem 10578 (Submitted solution.)
H. S. Wilf, Problem 10578, Amer. Math. Monthly, 104 (1997), 270.

Cf. A005171, A005450 (numerators).

Cf. A088140, A089026, A135684, A181569.

[IsPrime(n) select 1 else n: n in [1..70]]; // Vincenzo Librandi, Feb 22 2013

[Denominator((1 + Factorial(n-1))/n): n in [1..70]]; // G. C. Greubel, Nov 22 2022

seq(denom((1 + (n-1)!)/n), n=1..80); # G. C. Greubel, Nov 22 2022

Table[If[PrimeQ[n], 1, n], {n, 70}] (* Vincenzo Librandi, Feb 22 2013 *)
a[n_] := ((n-1)! + 1)/n - Floor[(n-1)!/n] // Denominator; Table[a[n] , {n, 70}] (* Jean-François Alcover, Jul 17 2013, after Minac's formula *)
Table[Denominator[(1 + Gamma[n])/n], {n,2,70}] (* G. C. Greubel, Nov 22 2022 *)

def A005451(n):
    if n == 4: return n
    f = factorial(n-1)
    return 1/((f + 1)/n - f//n)
[A005451(n) for n in (1..71)]   # Peter Luschny, Oct 16 2013

[denominator((1+gamma(n))/n) for n in range(1,71)] # G. C. Greubel, Nov 22 2022

Define b(n) = ( (n-1)*(n^2-3*n+1)*b(n-1) - (n-2)^3*b(n-2) )/(n*(n-3)); b(2) = b(3) = 1; a(n) = denominator(b(n)).
a(n) = A088140(n), n >= 3. - R. J. Mathar, Oct 28 2008
a(n) = gcd(n, (n!*n!!)/n^2). - Lechoslaw Ratajczak, Mar 09 2019
From Wesley Ivan Hurt, Jun 21 2025: (Start)
a(n) = n^c(n), where c = A005171.
a(n) = Sum_{d|n} A380441(d) * mu(n/d). (End)

Name edited and a(1)=1 prepended by G. C. Greubel, Nov 22 2022. Name further edited by N. J. A. Sloane, Nov 22 2022

A135683 Duplicate of A005451.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60, 1, 62, 63

Offset: 1

Mohammad K. Azarian, Dec 01 2007

dead

Previous name was: a(n) = 1 if n is a prime number, otherwise, a(n) = n.

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

[IsPrime(n) select 1 else n: n in [1..70]]; // Vincenzo Librandi, Feb 22 2013

seq(denom((1 + (n-1)!)/n), n=1..80); # G. C. Greubel, Nov 22 2022

Table[If[PrimeQ[n], 1, n], {n, 70}] (* Vincenzo Librandi, Feb 22 2013 *)
a[n_] := ((n-1)! + 1)/n - Floor[(n-1)!/n] // Denominator; Table[a[n] , {n, 1, 63}] (* Jean-François Alcover, Jul 17 2013, after Minac's formula *)

def A135683(n):
    if n == 4: return n
    f = factorial(n-1)
    return 1/((f + 1)/n - f//n)
[A135683(n) for n in (1..63)]   # Peter Luschny, Oct 16 2013

a(n) = A088140(n), n >= 3. - R. J. Mathar, Oct 28 2008
a(n) = gcd(n, (n!*n!!)/n^2). - Lechoslaw Ratajczak, Mar 09 2019
a(n) = A005451(n), for n >= 2. - G. C. Greubel, Nov 22 2022

A181569 Greatest common divisor of n! and n+1.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 8, 9, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60, 1, 62, 63, 64, 65, 66, 1, 68, 69, 70, 1, 72, 1, 74, 75, 76, 77, 78, 1

Offset: 1

Reinhard Zumkeller, Oct 31 2010

From Wilson's theorem, it follows that a(n) = 1 when n + 1 is prime, a(n) > 1 otherwise. - Alonso del Arte, Feb 25 2014

			a(6) = 1 because 6! and 7 are coprime.
a(7) = 8 because 7! = 5040 and gcd(5040, 8) = 8.
a(8) = 9 because 8! = 40320 and gcd(40320, 9) = 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000142, A006093, A010051, A050873.

Cf. A088140, A135683.

[GCD(Factorial(n), n+1): n in [1..80]]; // Vincenzo Librandi, Mar 03 2014

A181569:=n->gcd(n!,n+1): seq(A181569(n), n=1..100); # Wesley Ivan Hurt, Aug 13 2015

Table[GCD[n!, n + 1], {n, 80}] (* Alonso del Arte, Feb 25 2014 *)

a(n)= n!/denominator(polcoeff((x+1)*exp(x+x*O(x^n)), n));  \\ Gerry Martens, Aug 12 2015

A181569(n)=gcd(n!,n+1) \\ M. F. Hasler, Aug 16 2015

a(n) = A050873(A000142(n), n + 1);
a(A006093(n)) = 1;
for n > 3: a(n) = (n + 1) / (n*A010051(n+1) + 1).
a(n) = (n+1)/A014973(n+1). - Michel Marcus, Aug 14 2015

A088440 a(4n) = 4n, otherwise a(n) = 1.

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 12, 1, 1, 1, 16, 1, 1, 1, 20, 1, 1, 1, 24, 1, 1, 1, 28, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 40, 1, 1, 1, 44, 1, 1, 1, 48, 1, 1, 1, 52, 1, 1, 1, 56, 1, 1, 1, 60, 1, 1, 1, 64, 1, 1, 1, 68, 1, 1, 1, 72, 1, 1, 1, 76, 1, 1, 1, 80, 1, 1, 1, 84, 1, 1, 1, 88, 1, 1, 1, 92, 1, 1, 1

Offset: 0

Roger L. Bagula, Nov 09 2003

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A088140.

A088440:= func< n | n mod 4 eq 0 select n else 1>;
[A088440(n): n in [0..120]]; // G. C. Greubel, Dec 05 2022

Array[1 + (# - 1)*((1 + (-1)^(#/2))*(1 + (-1)^#))/4 &, 96, 0] (* Michael De Vlieger, May 07 2021 *)

A088440(n) = if((n%4),1,n); \\ Antti Karttunen, Jul 03 2018

def A088440(n): return 1 if (n%4) else n
[A088440(n) for n in range(121)] # G. C. Greubel, Dec 05 2022

a(n) = 1+(n-1)*((1+(-1)^(n/2))*(1+(-1)^n))/4. - Wesley Ivan Hurt, May 07 2021
G.f.: x*(1 + x + x^2 + 4*x^3 - x^4 - x^5 - x^6)/(1-x^4)^2. - Georg Fischer, Nov 17 2022
E.g.f.: (1/2)*( cosh(x) + (x+2)*sinh(x) - cos(x) - x*sin(x) ). - G. C. Greubel, Dec 05 2022

Description corrected by Antti Karttunen, Jul 03 2018

A088441 a(n) = n if n == 0 (mod 3), a(n) = 1 if n == 2 (mod 3), otherwise a(n) = floor((n-2)/2).

Original entry on oeis.org

1, 3, 1, 1, 6, 2, 1, 9, 4, 1, 12, 5, 1, 15, 7, 1, 18, 8, 1, 21, 10, 1, 24, 11, 1, 27, 13, 1, 30, 14, 1, 33, 16, 1, 36, 17, 1, 39, 19, 1, 42, 20, 1, 45, 22, 1, 48, 23, 1, 51, 25, 1, 54, 26, 1, 57, 28, 1, 60, 29, 1, 63, 31, 1, 66, 32, 1, 69, 34, 1, 72, 35, 1, 75, 37, 1, 78, 38, 1, 81, 40, 1

Offset: 2

Roger L. Bagula, Nov 09 2003

G. C. Greubel, Table of n, a(n) for n = 2..5002

Cf. A001651, A088140.

function A088441(n)
  if (n mod 3) eq 0 then return n;
  elif (n mod 3) eq 2 then return 1;
  else return Floor((n-2)/2);
  end if; return A088441;
end function;
[A088441(n): n in [2..100]]; // G. C. Greubel, Dec 05 2022

p[n_]= n!/Product[i, {i, n -Floor[2*n/3], n -Floor[n/3]}];
Table[Floor[p[n]/p[n-1]], {n,2,100}]
(* Second program *)
a[n_]:= If[Mod[n,3]==0, n, If[Mod[n,3]==2, 1, Floor[(n-2)/2]]];
Table[a[n], {n,2,100}] (* G. C. Greubel, Dec 05 2022 *)

def A088441(n):
    if (n%3)==0: return n
    elif (n%3)==2: return 1
    else: return (n-2)//2
[A088441(n) for n in range(2,100)] # G. C. Greubel, Dec 05 2022

a(n) = floor(p(n)/p(n-1)), where p(n) = n!/Product_{j=n-floor(2*n/3)..n-floor(n/3)} j.
From G. C. Greubel, Dec 05 2022: (Start)
a(n) = floor( n*Gamma(n - floor(2*n/3))*Gamma(n - floor((n-1)/3))/(Gamma(n - floor(n/3) + 1)*Gamma(n - floor(2*(n-1)/3) - 1)) ).
a(n) = n if n mod 3 = 0, 1 if n mod 3 = 2, otherwise floor((n-2)/2). (End)

Edited by G. C. Greubel, Dec 05 2022

A088494 Let P(n,k) = n!/(Product_{i=1..pi(n)/2^(k-1)} prime(i)) be an integer matrix of "partial" factorials. Then a(n) = sum_{k=1..8} floor( P(n,k)/P(n-1,k)).

Original entry on oeis.org

15, 20, 32, 36, 48, 41, 64, 72, 80, 78, 96, 81, 112, 120, 128, 120, 144, 94, 160, 168, 176, 162, 192, 200, 208, 216, 224, 177, 240, 218, 256, 264, 272, 280, 288, 195, 304, 312, 320, 288, 336, 261, 352, 360, 368, 330, 384, 392, 400, 408, 416, 212, 432, 440, 448

Offset: 2

Roger L. Bagula, Nov 10 2003

nonn

The auxiliary integer array P is n! divided by the product of the first primes with an upper limit of the prime index given by A000720(n)/2^(k-1). It starts in row n=1 with columns k>=1 as:
   1,   1,    1,    1,    1,    1,    1,    1, ...
   1,   2,    2,    2,    2,    2,    2,    2, ...
   1,   3,    6,    6,    6,    6,    6,    6, ...
   4,  12,   24,   24,   24,   24,   24,   24, ...
   4,  60,  120,  120,  120,  120,  120,  120, ...
  24, 360,  720,  720,  720,  720,  720,  720, ...
  24, 840, 2520, 5040, 5040, 5040, 5040, 5040, ...
The a(n) are some sort of average integer value of ratios of neighbored rows in the first 8 columns.

G. C. Greubel, Table of n, a(n) for n = 2..5000

Cf. A088140, A088493.

P := proc(n,k)
    local a,i ;
    a := 1 ;
    for i from 1 to numtheory[pi](n)/2^(k-1) do
        a := ithprime(i) *a ;
    end do:
    n!/a ;
end proc:
A088494 := proc(n)
    add( floor(P(n,k)/P(n-1,k)),k=1..8) ;
end proc: # R. J. Mathar, Sep 17 2013

p[n_, k_]:= p[n,k]= n!/Product[Prime[i], {i, PrimePi[n]/2^(k-1)}];
f[n_]:= f[n]= Sum[Floor[p[n, k]/p[n-1, k]], {k,8}];
Table[f[n], {n,2,70}]

@CachedFunction
def f(n,k): return product( nth_prime(j) for j in (1..prime_pi(n)/2^(k-1)) )
def A088494(n): return sum( (n*f(n-1,k)//f(n,k)) for k in (1..8) )
[A088494(n) for n in (2..70)] # G. C. Greubel, Mar 27 2022

a(n) = Sum_{k=1..8} floor(p(n,k)/p(n-1,k)), where p(n, k) = n!/( Product_{j=1..PrimePi(n)/2^(k-1)} Prime(j) ). - G. C. Greubel, Mar 27 2022

Meaningful name by R. J. Mathar, Sep 17 2013

A088438 A chaotic Cantor integer type product set of the factorial function that trifurcates.

Original entry on oeis.org

2, 6, 4, 7, 24, 35, 8, 18, 70, 88, 12, 29, 140, 165, 16, 40, 234, 266, 20, 52, 352, 391, 24, 64, 494, 540, 28, 76, 660, 713, 32, 88, 850, 910, 36, 99, 1064, 1131, 40, 111, 1302, 1376, 44, 123, 1564, 1645, 48, 135, 1850, 1938, 52, 147, 2160, 2255, 56, 159, 2494

Offset: 0

Roger L. Bagula, Nov 09 2003

This result is due to analysis of the prime product, composite product and factorial type function to a more general type of function: n!=Product[Set1[i],{i, limit1, limit2}]*Product[Set2[i],{i,limit3,limit4}] In this case the second product contains two intervals instead of one.

Cf. A088140.

(* factorial based function with half interval Cantor hole in the middle*) p[n_]=n!/Product[i, {i, n-Floor[n/4], n-Floor[3*n/4]}] digits=200 a0=Table[Floor[p[n]/p[n-1]], {n, 2, digits}]

P[n]=n!/Product[i, {i, n-Floor[n/4], n-Floor[3*n/4]}] a(n) = Floor[P[n]/P[n-1]]

cp's OEIS Frontend

A005451 a(n) = 1 if n is a prime number, otherwise a(n) = n.

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A135683 Duplicate of A005451.

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A181569 Greatest common divisor of n! and n+1.

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A088440 a(4n) = 4n, otherwise a(n) = 1.

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A088441 a(n) = n if n == 0 (mod 3), a(n) = 1 if n == 2 (mod 3), otherwise a(n) = floor((n-2)/2).

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A088494 Let P(n,k) = n!/(Product_{i=1..pi(n)/2^(k-1)} prime(i)) be an integer matrix of "partial" factorials. Then a(n) = sum_{k=1..8} floor( P(n,k)/P(n-1,k)).

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