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}.

1, 2, 3, 4, 8, 9, 16, 18, 20, 24, 25, 27, 28, 32, 56, 64, 81, 128, 162, 176, 192, 256, 352, 384, 736, 768, 896, 1026, 1029, 1056, 1280, 1792, 1863, 1864, 1928, 2052, 2058, 2064, 2080, 2304, 2432, 2560, 2944, 3776, 4376, 4384, 4480, 4482, 5104, 5120, 5121, 5125

Offset: 1

Author

Amiram Eldar, Mar 28 2025

Keywords

• nonn
• easy
• fini
• full

Comments

First differs from A382476 at n = 72: a(72) = 39936 while A382476(72) = 39937.

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..73
• Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
• Eric Weisstein's World of Mathematics, Niven's Constant.
• Wikipedia, Niven's constant.

Crossrefs

Cf. A033150, A051903, A129132, A380264, A380265, A382476.

Programs

Mathematica

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]

PARI

f(k) = if(k == 1, 0, vecmax(factor(k)[, 2]));
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);}