A060791 a(n) = n / gcd(n,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, 5, 26, 27, 28, 29, 6, 31, 32, 33, 34, 7, 36, 37, 38, 39, 8, 41, 42, 43, 44, 9, 46, 47, 48, 49, 10, 51, 52, 53, 54, 11, 56, 57, 58, 59, 12, 61, 62, 63, 64, 13, 66, 67, 68, 69

Offset: 1

Author

Len Smiley, Apr 26 2001

Keywords

• nonn
• mult
• easy

Comments

As well as being a multiplicative sequence, a(n) is also 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 20 2019

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Crossrefs

Cf. Sequences given by the formula n/gcd(n,k) = numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

Cf. A109046.

Programs

GAP

List([1..80],n->n/Gcd(n,5)); # Muniru A Asiru, Feb 20 2019

Magma

[n/GCD(n, 5): n in [1..100]]; // G. C. Greubel, Feb 20 2019

Maple

seq(n/gcd(n,5),n=1..80); # Muniru A Asiru, Feb 20 2019

Mathematica

f[n_]:=Numerator[n/(n+5)]; Array[f,100] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011*)

PARI

a(n) = { n / gcd(n, 5) } \\ Harry J. Smith, Jul 12 2009

PARI

a(n) = n/(5-4*(n%5>0)) \\ Zak Seidov, Feb 17 2011

Sage

[lcm(n,5)/5 for n in range(1, 51)] # Zerinvary Lajos, Jun 07 2009

Formula

G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + x^4 + 4*x^5 + 3*x^6 + 2*x^7 + x^8)/(1 - x^5)^2.

a(n) = n/5 if 5|n, otherwise a(n) = n.

From R. J. Mathar, Apr 18 2011: (Start)

a(n) = A109046(n)/5.

Dirichlet g.f.: zeta(s-1)*(1-4/5^s). (End)

G.f.: x*(x^4 + x^3 - x^2 + x + 1)*(x^4 + x^3 + 3*x^2 + x + 1)/((x - 1)^2*(x^4 + x^3 + x^2 + x + 1)^2). - R. J. Mathar, Oct 31 2015

From Peter Bala, Feb 20 2019: (Start)

a(n) = numerator(n/(n + 5)).

If gcd(n, m) = 1 then a(a(n)*a(m)) = a(a(n)) * a(a(m)), a(a(a(n))*a(a(m))) = a(a(a(n))) * a(a(a(m))) and so on.

G.f.: x/(1 - x)^2 - 4*x^5/(1 - x^5)^2. (End)

Sum_{k=1..n} a(k) ~ (21/50) * n^2. - Amiram Eldar, Nov 25 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = 9*log(2)/5. - Amiram Eldar, Sep 08 2023

Extensions

Extended (using terms from b-file) by Michel Marcus, Feb 08 2014