A377563 Numbers that have fewer infinitary divisors than noninfinitary divisors.

16, 36, 48, 80, 81, 100, 112, 144, 162, 176, 180, 196, 208, 225, 240, 252, 256, 272, 288, 300, 304, 324, 336, 368, 396, 400, 405, 432, 441, 450, 464, 468, 484, 496, 512, 528, 560, 567, 576, 588, 592, 612, 624, 625, 648, 656, 676, 684, 688, 700, 720, 752, 768, 784, 800

Offset: 1

Author

Amiram Eldar, Nov 01 2024

Keywords

• nonn
• easy

Comments

Numbers whose prime factorization has at least one exponent that has at least two zeros in its binary representation (A158582), or at least two exponents that are not of the form 2^k-1, with k >= 1 (A062289).

The asymptotic density of this sequence is 1 - d * (1 + Sum_{p prime} (Sum_{k>=0} 1/p^(3*2^k-1))/(1 + Sum_{k>=1} 1/p^(2^k-1))) = 0.07306380398261191432..., where d = A327839.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A037445, A062289, A077609, A083329, A158582, A327839, A335832, A348341, A377562.

Programs

Mathematica

f[p_, e_] := 2^DigitCount[e, 2, 1]/(e + 1); q[1] = False; q[n_] := Times @@ f @@@ FactorInteger[n] < 1/2; Select[Range[800], q]

PARI

is(n) = {my(f = factor(n)); vecprod(apply(x -> (1 << hammingweight(x)) / (x+1), f[, 2])) < 1/2;}