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.

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

Comments

First differs from A382475 at n = 72: a(72) = 39937 while A382475(72) = 39936.

Since lim_{k->oo} A129132(k)/k = c, this sequence is infinite if Niven's constant is irrational.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..103

Crossrefs

Cf. A033150, A051903, A129132, A380264, A380265, A382475.

Programs

Mathematica

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]

PARI

default(realprecision, 120);
f(k) = if(k == 1, 0, vecmax(factor(k)[, 2]));
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);}