A265204 Sum of phi(i) over squarefree numbers i <= n.

1, 2, 4, 4, 8, 10, 16, 16, 16, 20, 30, 30, 42, 48, 56, 56, 72, 72, 90, 90, 102, 112, 134, 134, 134, 146, 146, 146, 174, 182, 212, 212, 232, 248, 272, 272, 308, 326, 350, 350, 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

Offset: 1

Author

Jeffrey Shallit, Dec 04 2015

Keywords

• nonn

Comments

Partial sums of absolute values of A097945. - Robert Israel, Dec 10 2015

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A097945.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 0, a(n-1))+
      `if`(issqrfree(n), phi(n), 0)
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Dec 04 2015
N:= 1000: # to get a(1) to a(N)
V:= Vector(N, 1):
Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
for p in Primes do
  J1:= [seq(i,i=p..N,p)];
  J2:= [seq(i,i=p^2..N,p^2)];
  V[J1]:= V[J1] * (p-1);
  V[J2]:= 0;
od:
ListTools[PartialSums](convert(V,list)); # Robert Israel, Dec 10 2015

Mathematica

Table[Sum[EulerPhi@ i, {i, Select[Range@ n, SquareFreeQ]}], {n, 70}] (* Michael De Vlieger, Dec 10 2015 *)

PARI

a(n) = sum(i=1, n, eulerphi(i)*issquarefree(i)) \\ Anders Hellström, Dec 04 2015

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