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Least positive integer k such that Sum_{i=1..k} phi(i) >= 10^n.
1

%I #16 May 12 2021 22:49:03

%S 1,5,18,57,181,573,1814,5736,18138,57357,181380,573574,1813799,

%T 5735737,18137993,57357372,181379937,573573721,1813799364,5735737209,

%U 18137993642,57357372095,181379936423,573573720955,1813799364234,5735737209545,18137993642342,57357372095455

%N Least positive integer k such that Sum_{i=1..k} phi(i) >= 10^n.

%C Least integer k such that A002088(k) >= 10^n.

%C Since A002088(k) = (3*k^2)/(Pi^2) + O(k log k), the digits of a(n) for n even (resp. odd) approach the decimal digits of Pi/sqrt(3)=1.8137993642342... and Pi*sqrt(10/3)=5.7357372095454... resp.

%C Conjecture: For n even, either a(n) = floor(Pi/sqrt(3)*10^(n/2)) or a(n) = floor(Pi/sqrt(3)*10^(n/2))+1. For n odd, either a(n) = floor(Pi*sqrt(10/3)*10^((n-1)/2)) or a(n) = floor(Pi*sqrt(10/3)*10^((n-1)/2))+1.

%H Chai Wah Wu, <a href="/A343802/b343802.txt">Table of n, a(n) for n = 0..30</a>

%o (Python)

%o from sympy import totient

%o def A343802(n):

%o s, c = 0, 0

%o while s < 10**n:

%o c += 1

%o s += totient(c)

%o return c

%Y Cf. A002088, A038567.

%K nonn

%O 0,2

%A _Chai Wah Wu_, Apr 29 2021