A381070 Numbers k such that A380845(k)/k > A380845(m)/m for all m < k.

1, 2, 4, 8, 16, 18, 36, 72, 144, 288, 540, 1080, 2160, 4320, 8640, 17280, 34560, 45360, 68040, 90720, 106680, 136080, 213360, 272160, 320040, 640080, 1280160, 2560320, 2577960, 5155920, 10311840, 15467760, 30935520, 61871040, 123742080, 247484160, 494968320, 681080400

Offset: 1

Author

Amiram Eldar, Feb 13 2025

Keywords

• nonn
• base

Comments

Analogous to superabundant numbers (A004394) with A380845 instead of A000203.

The least number k for which A380845(k)/k >= 2 is k = a(6) = A380846(1) = 18.

The least number k for which A380845(k)/k >= 3 is k = a(12) = A380930(1) = 1080.

The least number k for which A380845(k)/k >= 4 is k = a(30) = A380931(1) = 5155920.

It seems that A380845(k)/k is unbounded (see the plot in the links section). What is the least number k for which A380845(k)/k >= 5?

Links

• Amiram Eldar, Table of n, a(n) for n = 1..46
• Amiram Eldar, Plot of A380845(n)/n at indices of record.

Crossrefs

Cf. A000203, A004394, A380845, A380846, A380929, A380930, A380931, A381069.

Programs

Mathematica

r[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, # &, DigitCount[#, 2, 1] == h &]/n]; seq[lim_] := Module[{s = {}, rm = 0, r1}, Do[r1 = r[k]; If[r1 > rm, rm = r1; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]

PARI

r(n) = {my(h = hammingweight(n)); sumdiv(n, d, d * (hammingweight(d) == h)) / n;}
list(lim) = {my(rm = 0, r1); for(k = 1, lim, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", "))); }