A346695 Numbers with more divisors than digits in their binary representation.

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 225, 228, 234, 240, 252, 260

Offset

1,1

Comments

Not all terms are perfect or abundant, with 105 being the first deficient term.

There are no primes in the sequence, and 6 is the only semiprime.

By the same comments as those at A175495, this sequence is infinite.

This sequence is a subsequence of A175495.

It is natural to conjecture that this sequence has asymptotic density 0. However, after the first three terms where a(n)/n = 6 -- a function which would increase to infinity if the asymptotic density were zero -- it drops, and it seems to take a long time to get that high again. The first time it gets above 5.0 is at a(30243)=151216. Even as high as a(2188516)=10000000, the density is only ~1/4.57.

The number of terms with m binary digits is Sum_{k>m} A346730(m,k). - Jon E. Schoenfield, Jul 31 2021

Links

Table of n, a(n) for n=1..59.

Example

12 has 6 divisors: {1,2,3,4,6,12}. 12 is written in binary as 1100, which has 4 digits. Since 6 > 4, 12 is in the sequence.

Mathematica

Select[Range[1000], (DivisorSigma[0, #] > Floor[1 + Log2[#]]) &]

Prog

(PARI) isok(m) = numdiv(m) > #binary(m); \\ Michel Marcus, Jul 29 2021
(Python)
from sympy import divisor_count
def ok(n): return divisor_count(n) > n.bit_length()
print(list(filter(ok, range(1, 261)))) # Michael S. Branicky, Jul 29 2021

Crossrefs

Cf. A000005, A070939.

Cf. A135772 (equal number rather than more).

Cf. A175495 (where "binary digits in n" is replaced by "log_2(n)").

Sequence in context: A051774 A119357 A097216 * A326133 A177052 A023196

Adjacent sequences: A346692 A346693 A346694 * A346696 A346697 A346698

Keyword

nonn,base,easy

Author

Alex Meiburg, Jul 29 2021

Status

approved