cp's OEIS Frontend

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

%I A064018 #76 Jun 25 2025 00:37:48
%S A064018 1,32,3044,304192,30397486,3039650754,303963552392,30396356427242,
%T A064018 3039635516365908,303963551173008414,30396355092886216366,
%U A064018 3039635509283386211140,303963550927059804025910,30396355092702898919527444,3039635509270144893910357854,303963550927013509478708835152
%N A064018 a(n) = A002088(10^n) = Sum_{k <= 10^n} phi(k), sum of the Euler totients phi = A000010.
%C A064018 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]
%H A064018 Lucas A. Brown, <a href="/A064018/b064018.txt">Table of n, a(n) for n = 0..19</a> (terms 0..18 from Hiroaki Yamanouchi)
%H A064018 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A064018.py">Python program</a>.
%H A064018 Lucas Augustus Brown, <a href="https://arxiv.org/abs/2506.07386">Computation of the Totient Summatory Function</a>, arXiv:2506.07386 [math.NT], 2025.
%H A064018 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientSummatoryFunction.html">Totient Summatory Function</a>.
%H A064018 Wikipedia, <a href="https://en.wikipedia.org/wiki/Totient_summatory_function">Totient summatory function</a>.
%F A064018 a(n) = Sum_{k <= 10^n} A000010(k).
%e A064018 a(1) = phi(1) + ... + phi(10) = 1 + 1 + 2 + 2 + 4 + 2 + 6 + 4 + 6 + 4 = 32.
%t A064018 s = 0; k = 1; Do[ While[ k <= 10^n, s = s + EulerPhi[ k ]; k++ ]; Print[ s ], {n, 0, 8} ]
%o A064018 (Python) # See LINKS. - _Lucas A. Brown_, Jun 08 2025
%Y A064018 Cf. A000010, A002088, A104141.
%K A064018 nonn
%O A064018 0,2
%A A064018 _Robert G. Wilson v_, Sep 07 2001
%E A064018 More terms from _Robert G. Wilson v_, Sep 07 2001
%E A064018 a(10)-a(11) from _Donovan Johnson_, Feb 06 2010
%E A064018 a(12) from _Donovan Johnson_, Feb 07 2012
%E A064018 a(13)-a(14) from _Hiroaki Yamanouchi_, Jul 06 2014
%E A064018 a(15) from _Asif Ahmed_, Apr 16 2015
%E A064018 Name edited by _Michel Marcus_ and _M. F. Hasler_, Apr 16 and Apr 18 2015