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A385561 Numbers m such that (1/m) * Sum_{k=1..m} phi(k)/k is closer to 6/Pi^2 than it is for any number smaller than m, where phi is the Euler totient function (A000010).

%I A385561 #6 Jul 05 2025 11:24:59
%S A385561 1,2,3,4,6,10,12,16,22,28,36,66,96,100,126,156,190,330,430,540,820,
%T A385561 876,1086,1422,10596,10836,18096,35796,55786,69336,111100,168666,
%U A385561 284650,905950,1482300,1745590,2405560,2661310,4023306,5869956,17454580,25670646,51305346,79969618,211025650,622626790
%N A385561 Numbers m such that (1/m) * Sum_{k=1..m} phi(k)/k is closer to 6/Pi^2 than it is for any number smaller than m, where phi is the Euler totient function (A000010).
%C A385561 6/Pi^2 is the asymptotic mean of phi(k)/k, i.e., lim_{m->oo} (1/m) * Sum_{k=1..m} phi(k)/k = 6/Pi^2 (Walfisz, 1963; Sándor et al., 2005).
%D A385561 József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 27.
%D A385561 Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Berlin, 1963.
%t A385561 seq[lim_] := Module[{s = {}, sum = 0, dm = 1, d}, Do[sum += EulerPhi[k]/k; If[(d = Abs[sum/k - 6/Pi^2]) < dm, dm = d; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]
%o A385561 (PARI) list(lim) = {my(sm = 0, dm = 1, d); for(k = 1, lim, sm += eulerphi(k)/k; d = abs(sm/k - 6/Pi^2); if(d < dm, dm = d; print1(k, ", ")));}
%Y A385561 Cf. A000010, A059956, A071708, A072155, A330899, A385562.
%K A385561 nonn
%O A385561 1,2
%A A385561 _Amiram Eldar_, Jul 03 2025