A064018 a(n) = A002088(10^n) = Sum_{k <= 10^n} phi(k), sum of the Euler totients phi = A000010.

1, 32, 3044, 304192, 30397486, 3039650754, 303963552392, 30396356427242, 3039635516365908, 303963551173008414, 30396355092886216366, 3039635509283386211140, 303963550927059804025910, 30396355092702898919527444, 3039635509270144893910357854, 303963550927013509478708835152

Offset: 0

Author

Robert G. Wilson v, Sep 07 2001

Keywords

• nonn

Comments

Asymptotically, A002088(n) ~ 0.30396355...*n^2 = (3/Pi^2)*n^2, see A104141 and A002088. - Michael B. Porter, Mar 08 2013 [corrected by M. F. Hasler, Apr 18 2015]

Examples

			a(1) = phi(1) + ... + phi(10) = 1 + 1 + 2 + 2 + 4 + 2 + 6 + 4 + 6 + 4 = 32.
		

Links

• Lucas A. Brown, Table of n, a(n) for n = 0..19 (terms 0..18 from Hiroaki Yamanouchi)
• Lucas A. Brown, Python program.
• Lucas Augustus Brown, Computation of the Totient Summatory Function, arXiv:2506.07386 [math.NT], 2025.
• Eric Weisstein's World of Mathematics, Totient Summatory Function.
• Wikipedia, Totient summatory function.

Crossrefs

Cf. A000010, A002088, A104141.

Programs

Mathematica

s = 0; k = 1; Do[ While[ k <= 10^n, s = s + EulerPhi[ k ]; k++ ]; Print[ s ], {n, 0, 8} ]

Python

# See LINKS. - Lucas A. Brown, Jun 08 2025

Formula

a(n) = Sum_{k <= 10^n} A000010(k).

Extensions

More terms from Robert G. Wilson v, Sep 07 2001

a(10)-a(11) from Donovan Johnson, Feb 06 2010

a(12) from Donovan Johnson, Feb 07 2012

a(13)-a(14) from Hiroaki Yamanouchi, Jul 06 2014

a(15) from Asif Ahmed, Apr 16 2015

Name edited by Michel Marcus and M. F. Hasler, Apr 16 and Apr 18 2015