A060791 - OEIS

%I #54 Sep 08 2023 07:33:01

%S 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,

%T 28,29,6,31,32,33,34,7,36,37,38,39,8,41,42,43,44,9,46,47,48,49,10,51,

%U 52,53,54,11,56,57,58,59,12,61,62,63,64,13,66,67,68,69

%N a(n) = n / gcd(n,5).

%C 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

%H Harry J. Smith, <a href="/A060791/b060791.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,2,0,0,0,0,-1).

%F 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.

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

%F From _R. J. Mathar_, Apr 18 2011: (Start)

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

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

%F 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

%F From _Peter Bala_, Feb 20 2019: (Start)

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

%F 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.

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

%F Sum_{k=1..n} a(k) ~ (21/50) * n^2. - _Amiram Eldar_, Nov 25 2022

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

%p seq(n/gcd(n,5),n=1..80); # _Muniru A Asiru_, Feb 20 2019

%t f[n_]:=Numerator[n/(n+5)]; Array[f,100] (* _Vladimir Joseph Stephan Orlovsky_, Feb 16 2011*)

%o (Sage) [lcm(n,5)/5 for n in range(1, 51)] # _Zerinvary Lajos_, Jun 07 2009

%o (PARI) { for (n=1, 1000, write("b060791.txt", n, " ", n / gcd(n, 5)); ) } \\ _Harry J. Smith_, Jul 12 2009

%o (PARI) for (n=1,1000,print1(n/(5-4*(n%5>0))", ")) \\ _Zak Seidov_, Feb 17 2011

%o (Magma) [n/GCD(n, 5): n in [1..100]]; // _G. C. Greubel_, Feb 20 2019

%o (GAP) List([1..80],n->n/Gcd(n,5)); # _Muniru A Asiru_, Feb 20 2019

%Y 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).

%Y Cf. A109046.

%K nonn,mult,easy

%O 1,2

%A _Len Smiley_, Apr 26 2001

%E Extended (using terms from b-file) by _Michel Marcus_, Feb 08 2014