A354768 Numbers k such that d(k)/k >= d(m)/m for all m > k, where d(k) is the number-of-divisors function A000005(k).

1, 2, 4, 6, 8, 12, 18, 24, 30, 36, 48, 60, 72, 84, 90, 120, 144, 180, 240, 252, 360, 420, 480, 504, 540, 720, 840, 900, 1008, 1080, 1260, 1440, 1680, 1800, 2520, 2640, 2880, 3360, 3780, 3960, 5040, 5280, 5400, 5460, 5544, 6300, 7560, 7920, 8400, 10080, 10920, 12600, 15120, 15840, 16380, 18480

Offset

1,2

Comments

Because of the bound d(m) <= 2*sqrt(m), in order for k to be in the sequence it suffices that d(k)/k >= d(m)/m for k < m < (2*k/d(k))^2. - Robert Israel, Jan 23 2023

References

David desJardins, Posting to Math Fun Mailing List, Jun 21 2022.

Links

Table of n, a(n) for n=1..56.

Maple

N:= 10^6:
Q:= [seq(numtheory:-tau(k)/k, k=1..N)]:
V:= Vector(10^6):
r:= 2/10^3:
for n from 10^6 to 1 by -1 do
r:= max(Q[n], r);
V[n]:= r;
od:
select(i -> Q[i] >= V[i+1], [$1..10^6-1]); # Robert Israel, Jan 23 2023

Crossrefs

Cf. A000005, A066523, A354769, A368523.

Sequence in context: A032458 A284005 A079212 * A141452 A292854 A330813

Adjacent sequences: A354765 A354766 A354767 * A354769 A354770 A354771

Keyword

nonn

Author

N. J. A. Sloane, Jun 21 2022

Extensions

More terms from Robert Israel, Jan 23 2023

Status

approved