A069193 - OEIS

%I #23 Sep 21 2023 01:45:33

%S 1,3,5,10,9,15,13,28,24,27,21,50,25,39,45,72,33,72,37,90,65,63,45,140,

%T 70,75,99,130,57,135,61,176,105,99,117,240,73,111,125,252,81,195,85,

%U 210,216,135,93,360,140,210,165,250,105,297,189,364,185,171,117,450

%N a(n) = Sum_{d|n} d*phi(n)/phi(d).

%H Amiram Eldar, <a href="/A069193/b069193.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = ((e+1)*p - 1) * p^(e-1). - _Amiram Eldar_, Sep 15 2019

%F a(n) = Sum_{k=1..n, gcd(n,k) = 1} sigma(gcd(n,k-1)). - _Ilya Gutkovskiy_, Sep 24 2021

%F Dirichlet g.f.: zeta(s-1)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(2*s-1)). - _Amiram Eldar_, Sep 21 2023

%p with(numtheory):for n from 1 to 200 do di := sort(convert(divisors(n),list)): a[n] := sum(di[i]*phi(n)/phi(di[i]),i=1..nops(di)):od:seq(a[k],k=1..200);

%t f[p_, e_] := ((e+1)*p - 1) * p^(e-1) ; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 15 2019 *)

%o (PARI) a(n) = sumdiv(n, d, d*eulerphi(n)/eulerphi(d)); \\ _Michel Marcus_, Sep 15 2019

%o (Magma) [&+[d*EulerPhi(n) div EulerPhi(d):d in Divisors(n)]:n in [1..60]]; // _Marius A. Burtea_, Sep 15 2019

%Y Cf. A000010.

%K mult,easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Apr 10 2002

%E More terms from _Sascha Kurz_, Feb 02 2003