A069087 - OEIS

A069087

Numbers m such that (1/m)*Sum_{k=1..m} core(k) > phi(m) where core(n) = A007913(n) is the squarefree part of n: the smallest number such that n*a(n) is a square and phi(n) = A000010(n) is the Euler totient function.

%I #20 Apr 02 2020 02:51:44

%S 2,6,12,18,24,30,36,42,48,60,66,72,78,84,90,96,102,114,120,126,132,

%T 138,144,150,156,168,174,180,186,198,204,210,222,228,234,240,246,252,

%U 258,264,270,276,282,294,300,306,312,318,330,336,342,348,360,372,378,390

%N Numbers m such that (1/m)*Sum_{k=1..m} core(k) > phi(m) where core(n) = A007913(n) is the squarefree part of n: the smallest number such that n*a(n) is a square and phi(n) = A000010(n) is the Euler totient function.

%C Equivalently, numbers m such that A069891(m) > m*phi(m).

%C The listed terms are all even, but there are some odd terms, including m = 111546435 = 3*5*7*11*13*17*19*23, for which A069891(m) = 4093453424286382 and m*phi(m) = 4070927302041600.

%H Amiram Eldar, <a href="/A069087/b069087.txt">Table of n, a(n) for n = 1..10000</a>

%t f[p_, e_] := If[OddQ[e], p, 1]; sqf[n_] := Times @@ (f @@@ FactorInteger[n]); seq = {}; s = 0; Do[s += sqf[n]; If[s > n*EulerPhi[n], AppendTo[seq, n]], {n, 1, 400}]; seq (* _Amiram Eldar_, Apr 02 2020 *)

%o (PARI) is(n)=sum(k=1,n,core(k)) > n*eulerphi(n) \\ _Charles R Greathouse IV_, Feb 21 2013

%K nonn

%O 1,1

%A _Benoit Cloitre_, Apr 05 2002

%E Edited by _Dean Hickerson_, Apr 09 2002