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

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Comments

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

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

References

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

Links

• 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.

Crossrefs

Cf. A005171, A005450 (numerators).

Cf. A088140, A089026, A135684, A181569.

Programs

Magma

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

Magma

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

Maple

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

Mathematica

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 *)

Sage

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

SageMath

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

Formula

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)

Extensions

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