A360641 Numbers k where A093653(k)/A000120(k) sets a new record.

1, 2, 4, 8, 12, 16, 24, 36, 66, 72, 132, 144, 264, 420, 528, 840, 1026, 1056, 1680, 2052, 4104, 8208, 16416, 32832, 65664, 73920, 84000, 110880, 118800, 131328, 133380, 237600, 263340, 266760, 526680, 533520, 1053360, 1067040, 2106720, 2134080, 3160080, 4213440

Offset: 1

Author

Amiram Eldar, Feb 15 2023

Keywords

• nonn
• base

Comments

Analogous to superabundant numbers (A004394) as A175526 is analogous to abundant numbers (A005101).

The corresponding record values are 1, 2, 3, 4, 9/2, 5, 6, 15/2, 8, 10, ... .

This sequence is infinite since A093653(k)/A000120(k) is unbounded: A093653(2^m)/A000120(2^m) = m+1 for all m >= 0.

Examples

			The values of A093653(k)/A000120(k) for k=1..10 are 1, 2, 3/2, 3, 3/2, 3, 4/3, 4, 5/2 and 3. The record values, 1, 2, 3 and 4, occur at 1, 2, 4 and 8, the first 4 terms of this sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..58 (terms below 10^10)

Crossrefs

Cf. A000120, A004394, A005101, A093653, A093687, A175522, A175526, A360642.

Programs

Mathematica

seq[nmax_] := Module[{s = {}, rm = 0, r}, Do[If[(r = DivisorSum[n, DigitCount[#, 2, 1] &]/DigitCount[n, 2, 1]) > rm, rm = r; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[10^4]

PARI

lista(kmax) = {my(rm = 0, r); for(k = 1, kmax, r = sumdiv(k, d, hammingweight(d))/hammingweight(k); if(r > rm, rm = r; print1(k, ", "))); }

Python

# uses imports and definitions in A093653, A000120
from itertools import count, islice
def f(n): return A093653(n)/A000120(n)
def agen(r=0): yield from ((m, r:=fm)[0] for m in count(1) if (fm:=f(m))>r)
print(list(islice(agen(), 34))) # Michael S. Branicky, Feb 15 2023