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A382475 Numbers k where record values occur for A129132(k)/k = A380264(k)/A380265(k), the mean value of the maximum exponent in the prime factorization of the numbers {1, 2, ..., k}.

%I A382475 #10 Mar 28 2025 08:00:15
%S A382475 1,2,3,4,8,9,16,18,20,24,25,27,28,32,56,64,81,128,162,176,192,256,352,
%T A382475 384,736,768,896,1026,1029,1056,1280,1792,1863,1864,1928,2052,2058,
%U A382475 2064,2080,2304,2432,2560,2944,3776,4376,4384,4480,4482,5104,5120,5121,5125
%N A382475 Numbers k where record values occur for A129132(k)/k = A380264(k)/A380265(k), the mean value of the maximum exponent in the prime factorization of the numbers {1, 2, ..., k}.
%C A382475 First differs from A382476 at n = 72: a(72) = 39936 while A382476(72) = 39937.
%C A382475 Niven (1969) proved that abs(A129132(k)/k - c) < f(k) = (3/k) * Sum_{i=2 .. floor(log_2(k))} k^(1/i), where c = A033150 is Niven's constant. For k = 81984 we have A129132(k)/k - c = 2.40277...*10^(-5). There are no other terms in this sequence that are larger than 81984 up to 16500000000, and for k = 16500000000 we have abs(A129132(k)/k - c) < f(k) = 2.39403...*10^(-5). Therefore, this sequence is finite and a(73) = 81984 is the last term of this sequence.
%H A382475 Amiram Eldar, <a href="/A382475/b382475.txt">Table of n, a(n) for n = 1..73</a>
%H A382475 Ivan Niven, <a href="https://doi.org/10.1090/S0002-9939-1969-0241373-5">Averages of Exponents in Factoring Integers</a>, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
%H A382475 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NivensConstant.html">Niven's Constant</a>.
%H A382475 Wikipedia, <a href="https://en.wikipedia.org/wiki/Niven%27s_constant">Niven's constant</a>.
%t A382475 f[k_] := Max[FactorInteger[k][[;; , 2]]]; f[1] = 0; seq[lim_] := Module[{v = {}, s = 0, rm = -1, r}, Do[s += f[k]; r = s/k; If[r > rm, rm = r; AppendTo[v, k]], {k, 1, lim}]; v]; seq[10^5]
%o A382475 (PARI) f(k) = if(k == 1, 0, vecmax(factor(k)[, 2]));
%o A382475 list(lim) = {my(v = List(), s = 0, rm = -1, r); for(k = 1, lim, s += f(k); r = s/k; if(r > rm, rm = r; listput(v, k))); Vec(v);}
%Y A382475 Cf. A033150, A051903, A129132, A380264, A380265, A382476.
%K A382475 nonn,easy,fini,full
%O A382475 1,2
%A A382475 _Amiram Eldar_, Mar 28 2025