A359101 - OEIS

%I #20 Dec 17 2022 03:13:47

%S 1,1,2,2,5,2,6,4,6,5,10,4,12,6,10,8,16,6,18,10,12,10,22,8,25,12,18,12,

%T 28,10,30,16,20,16,30,12,36,18,24,20,40,12,42,20,30,22,46,16,42,25,32,

%U 24,52,18,50,24,36,28,58,20,60,30,36,32,60,20,66,32,44,30,70,24,72,36,50,36,60,24,78

%N a(n) = phi(5 * n)/4.

%H Michael De Vlieger, <a href="/A359101/b359101.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MoebiusTransform.html">Moebius Transform</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.

%F G.f.: -Sum_{k>=1} mu(5 * k) * x^k / (1 - x^k)^2, where mu() is the Moebius function (A008683).

%F From _Amiram Eldar_, Dec 17 2022: (Start)

%F Multiplicative with a(5^e) = 5^e, and a(p^e) = (p-1)*p^(e-1) if p != 5.

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

%F Sum_{k=1..n} a(k) ~ (25/(8*Pi^2)) * n^2. (End)

%t Array[EulerPhi[5 #]/4 &, 79] (* _Michael De Vlieger_, Dec 16 2022 *)

%o (PARI) a(n) = eulerphi(5*n)/4;

%o (PARI) my(N=80, x='x+O('x^N)); Vec(-sum(k=1, N, moebius(5*k)*x^k/(1-x^k)^2))

%Y Cf. A195459, A359102.

%Y Cf. A008653, A359100.

%K nonn,easy,mult

%O 1,3

%A _Seiichi Manyama_, Dec 16 2022