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

1,1

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

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]

Prog

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

Crossrefs

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

Sequence in context: A334538 A066112 A223456 * A103843 A347747 A390236

Adjacent sequences: A377560 A377561 A377562 * A377564 A377565 A377566

Keyword

nonn,easy

Author

Amiram Eldar, Nov 01 2024

Status

approved