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A382476 Numbers k where record low values occur for abs(A129132(k)/k - c) = abs(A380264(k)/A380265(k) - c), where c = A033150 is Niven's constant.

%I A382476 #8 Mar 28 2025 08:00:07
%S A382476 1,2,3,4,8,9,16,18,20,24,25,27,28,32,56,64,81,128,162,176,192,256,352,
%T A382476 384,736,768,896,1026,1029,1056,1280,1792,1863,1864,1928,2052,2058,
%U A382476 2064,2080,2304,2432,2560,2944,3776,4376,4384,4480,4482,5104,5120,5121,5125
%N A382476 Numbers k where record low values occur for abs(A129132(k)/k - c) = abs(A380264(k)/A380265(k) - c), where c = A033150 is Niven's constant.
%C A382476 First differs from A382475 at n = 72: a(72) = 39937 while A382475(72) = 39936.
%C A382476 Since lim_{k->oo} A129132(k)/k = c, this sequence is infinite if Niven's constant is irrational.
%H A382476 Amiram Eldar, <a href="/A382476/b382476.txt">Table of n, a(n) for n = 1..103</a>
%t A382476 f[k_] := Max[FactorInteger[k][[;; , 2]]]; f[1] = 0; seq[lim_] := Module[{Niven = 1 + NSum[1 - 1/Zeta[k], {k, 2, Infinity}, NSumTerms -> 100, WorkingPrecision -> 50], dm = 2, d, s = {}, sum = 0}, Do[sum += f[k]; d = Abs[sum/k - Niven]; If[d < dm, dm = d; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^4]
%o A382476 (PARI) default(realprecision, 120);
%o A382476 f(k) = if(k == 1, 0, vecmax(factor(k)[, 2]));
%o A382476 list(lim) = {my(niven = 1 + suminf(k = 2, 1-1/zeta(k)), dm = 2, d, s = List(), sm = 0); for(k = 1, lim, sm += f(k); d = abs(sm/k - niven); if(d < dm, dm = d; listput(s, k))); Vec(s);}
%Y A382476 Cf. A033150, A051903, A129132, A380264, A380265, A382475.
%K A382476 nonn
%O A382476 1,2
%A A382476 _Amiram Eldar_, Mar 28 2025