A265204 - OEIS

%I #26 Dec 24 2015 10:18:53

%S 1,2,4,4,8,10,16,16,16,20,30,30,42,48,56,56,72,72,90,90,102,112,134,

%T 134,134,146,146,146,174,182,212,212,232,248,272,272,308,326,350,350,

%U 390,402,444,444,444,466,512,512,512,512,544,544,596,596,636,636,672,700,758,758,818,848,848,848,896,916,982,982,1026,1050

%N Sum of phi(i) over squarefree numbers i <= n.

%C Partial sums of absolute values of A097945. - _Robert Israel_, Dec 10 2015

%H Robert Israel, <a href="/A265204/b265204.txt">Table of n, a(n) for n = 1..10000</a>

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 0, a(n-1))+

%p       `if`(issqrfree(n), phi(n), 0)

%p     end:

%p seq(a(n), n=1..70);  # _Alois P. Heinz_, Dec 04 2015

%p N:= 1000: # to get a(1) to a(N)

%p V:= Vector(N, 1):

%p Primes:= select(isprime, [2,seq(i,i=3..N,2)]):

%p for p in Primes do

%p   J1:= [seq(i,i=p..N,p)];

%p   J2:= [seq(i,i=p^2..N,p^2)];

%p   V[J1]:= V[J1] * (p-1);

%p   V[J2]:= 0;

%p od:

%p ListTools[PartialSums](convert(V,list)); # _Robert Israel_, Dec 10 2015

%t Table[Sum[EulerPhi@ i, {i, Select[Range@ n, SquareFreeQ]}], {n, 70}] (* _Michael De Vlieger_, Dec 10 2015 *)

%o (PARI) a(n) = sum(i=1, n, eulerphi(i)*issquarefree(i)) \\ _Anders Hellström_, Dec 04 2015

%o (Perl) use ntheory ":all"; sub an { vecsum(map { is_square_free($_) ? euler_phi($_) : () } 1..shift); } say an($_) for 1..70; # _Dana Jacobsen_, Dec 10 2015

%Y Cf. A000010, A097945.

%K nonn

%O 1,2

%A _Jeffrey Shallit_, Dec 04 2015