A132739 Largest divisor of n not divisible by 5.

1, 2, 3, 4, 1, 6, 7, 8, 9, 2, 11, 12, 13, 14, 3, 16, 17, 18, 19, 4, 21, 22, 23, 24, 1, 26, 27, 28, 29, 6, 31, 32, 33, 34, 7, 36, 37, 38, 39, 8, 41, 42, 43, 44, 9, 46, 47, 48, 49, 2, 51, 52, 53, 54, 11, 56, 57, 58, 59, 12, 61, 62, 63, 64, 13, 66, 67, 68, 69, 14, 71, 72, 73, 74, 3, 76, 77

Offset

1,2

Comments

A000265(a(n)) = a(A000265(n)) = A132740(n).

a(n) = A060791(n) when n is not divisible by 5. When n is divisible by 5 a(n) divides A060791(n). Tom Edgar, Feb 08 2014

As well as being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 21 2019

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Peter Bala, A note on the sequence of numerators of a rational function, 2019.

Formula

a(n) = n/A060904(n). Dirichlet g.f.: zeta(s-1)*(5^s-5)/(5^s-1). - R. J. Mathar, Jul 12 2012

a(n) = n/5^A112765(n). See A060904. - Wolfdieter Lang, Jun 18 2014

From Peter Bala, Feb 21 2019: (Start)

a(n) = n/gcd(n,5^n).

O.g.f.: F(x) - 4*F(x^5) - 4*F(x^25) - 4*F(x^125) - ..., where F(x) = x/(1 - x)^2 is the generating function for the positive integers. More generally, for m >= 1,

Sum_{n >= 0} a(n)^m*x^n = F(m,x) - (5^m - 1)(F(m,x^5) + F(m,x^25) + F(m,x^125) + ...), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m_th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.

Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence produces generating functions for the sequences n^m*a(n), m in Z. Some examples are given below. (End)

Sum_{k=1..n} a(k) ~ (5/12) * n^2. - Amiram Eldar, Nov 28 2022

Example

From Peter Bala, Feb 21 2019: (Start)
Sum_{n >= 1} n*a(n)*x^n = G(x) - (4*5)*G(x^5) - (4*25)*G(x^25) - (4*125)*G(x^125) - ..., where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (4/5)*H(x^5) - (4/25)*H(x^25) - (4/125)*H(x^125) - ..., where H(x) = x/(1 - x).
Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (4/5^2)*L(x^5) - (4/25^2)*L(x^25) - (4/125^2)*L(x^125) - ..., where L(x) = Log(1/(1 - x)).
Also, Sum_{n >= 1} 1/a(n)*x^n = L(x) + (4/5)*L(x^5) + (4/5)*L(x^25) + (4/5)*L(x^125) + ....
(End)

Mathematica

f[n_]:=Denominator[5^n/n]; Array[f, 100] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011*)
Table[n/5^IntegerExponent[n, 5], {n, 100}] (* Amiram Eldar, Sep 15 2020 *)

Prog

(Haskell)
a132739 n | r > 0     = n
          | otherwise = a132739 n' where (n', r) = divMod n 5
-- Reinhard Zumkeller, Apr 08 2011
(PARI) a(n)=n/5^valuation(n, 5) /* Simon Strandgaard, Nov 01 2021 */
(Ruby) p (1..50).map { |n| n /= 5 while (n % 5) == 0; n } # Simon Strandgaard, Nov 01 2021
(Python)
def A132739(n):
    a, b = divmod(n, 5)
    while b == 0:
        a, b = divmod(a, 5)
    return 5*a+b # Chai Wah Wu, Dec 05 2021

Crossrefs

Cf. A000265, A038502, A060791, A060904, A112765, A242603.

Sequence in context: A277826 A319653 A072438 * A060791 A235380 A116912

Adjacent sequences: A132736 A132737 A132738 * A132740 A132741 A132742

Keyword

nonn,mult

Author

Reinhard Zumkeller, Aug 27 2007

Status

approved