OFFSET
1,2
COMMENTS
That is, if the divisors of a number are listed in increasing order, the ratio of adjacent divisors is at most 2. The only odd number in this sequence is 1. Every term appears to be a practical number (A005153). The first practical number not here is 78.
Let p1^e1 * p2^e2 * ... * pr^er be the prime factorization of a number, with primes p1 < p2 < ... < pr and ek > 0. Then the number is in this sequence if and only if pk <= 2*Product_{j < k} p_j^e_j. This condition is similar to, but more restrictive than, the condition for practical numbers.
The polymath8 project led by Terry Tao refers to these numbers as "2-densely divisible". In general they say that n is y-densely divisible if its divisors increase by a factor of y or less, or equivalently, if for every R with 1 <= R <= n, there is a divisor in the interval [R/y,R]. They use this as a weakening of the condition that n be y-smooth. - David S. Metzler, Jul 02 2013
Is this the same as numbers k with the property that the symmetric representation of sigma(k) has only one part? If not, where is the first place these sequences differ? (cf. A237593). - Omar E. Pol, Mar 06 2014
Yes, the sequence so defined is the same as this sequence; see proof in the links. - Hartmut F. W. Hoft, Nov 26 2014
Saias (1997) called these terms "2-dense numbers" and proved that if N(x) is the number of terms not exceeding x, then there are two positive constants c_1 and c_2 such that c_1 * x/log_2(x) <= N(x) <= c_2 * x/log_2(x) for all x >= 2. - Amiram Eldar, Jul 23 2020
Weingartner (2015, 2019) showed that N(x) = c*x/log(x) + O(x/(log(x))^2), where c = 1.224830... As a result, a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 0.816439... - Andreas Weingartner, Jun 22 2021
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
José Manuel Rodríguez Caballero, Jordan's Expansion of the Reciprocal of Theta Functions and 2-densely Divisible Numbers, Integers, Vol. 20 (2020), Article A2.
Hartmut F. W. Hoft, Proof of a conjecture.
Hartmut F. W. Hoft, Proof of a second conjecture.
Eric Saias, Entiers à diviseurs denses 1, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.
Terence Tao, A Truncated Elementary Selberg Sieve of Pintz. (blog entry defining y-densely divisible)
Terence Tao et al., Polymath8 home page.
Andreas Weingartner, Practical numbers and the distribution of divisors, The Quarterly Journal of Mathematics, Vol. 66, No. 2 (2015), pp. 743-758; arXiv preprint, arXiv:1405.2585 [math.NT], 2014-2015.
Andreas Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp., Vol. 88, No. 318 (2019), pp. 1883-1902; arXiv preprint, arXiv:1705.06349 [math.NT], 2017-2018.
Andreas Weingartner, The number of prime factors of integers with dense divisors, arXiv:2101.11585 [math.NT], 2021.
Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, arXiv:2303.16819 [math.NT], 2023.
FORMULA
a(n) = A047836(n) / 2. - Reinhard Zumkeller, Sep 28 2011
a(n) = C*n*log(n*log(n)) + O(n), where C = 0.816439... (see comments). - Andreas Weingartner, Jun 23 2021
EXAMPLE
The divisors of 12 are 1, 2, 3, 4, 6, 12. The ratios of adjacent divisors is 2, 3/2, 4/3, 3/2, and 2, which are all <= 2. Hence 12 is in this sequence.
Example from Omar E. Pol, Mar 06 2014: (Start)
The symmetric representation of sigma(6) = 12 in the first quadrant looks like this:
y
.
._ _ _ _
|_ _ _ |_
. | |_
. |_ _ |
. | |
. | |
. . . . . |_| . . x
.
6 is in the sequence because the symmetric representation of sigma(6) = 12 has only one part. The 6th row of A237593 is [4, 1, 1, 1, 1, 4] and the 5th row of A237593 is [3, 2, 2, 3] therefore between both symmetric Dyck paths there is only one region (or part) of size 12.
70 is not in the sequence because the symmetric representation of sigma(70) = 144 has three parts. The 70th row of A237593 is [36, 12, 6, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 6, 12, 36] and the 69th row of A237593 is [35, 12, 7, 4, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 7, 12, 35] therefore between both symmetric Dyck paths there are three regions (or parts) of size [54, 36, 54]. (End)
MAPLE
a:= proc() option remember; local k; for k from 1+`if`(n=1, 0,
a(n-1)) while (l-> ormap(x-> x, [seq(l[i]>l[i-1]*2, i=2..
nops(l))]))(sort([(numtheory[divisors](k))[]])) do od; k
end:
seq(a(n), n=1..100); # Alois P. Heinz, Jul 27 2018
MATHEMATICA
OK[n_] := Module[{d=Divisors[n]}, And@@(#<=2& /@ (Rest[d]/Most[d]))]; Select[Range[1000], OK]
dif2Q[n_]:=AllTrue[#[[2]]/#[[1]]&/@Partition[Divisors[n], 2, 1], #<=2&]; Select[Range[300], dif2Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 29 2020 *)
PROG
(Haskell)
a174973 n = a174973_list !! (n-1)
a174973_list = filter f [1..] where
f n = all (<= 0) $ zipWith (-) (tail divs) (map (* 2) divs)
where divs = a027750_row' n
-- Reinhard Zumkeller, Jun 25 2015, Sep 28 2011
(PARI) is(n)=my(d=divisors(n)); for(i=2, #d, if(d[i]>2*d[i-1], return(0))); 1 \\ Charles R Greathouse IV, Jul 06 2013
(Magma) [k:k in [1..260]|forall{i:i in [1..#Divisors(k)-1]|d[i+1]/d[i] le 2 where d is Divisors(k)}]; // Marius A. Burtea, Jan 09 2020
(Python)
from sympy import divisors
def ok(n):
d = divisors(n)
return all(d[i]/d[i-1] <= 2 for i in range(1, len(d)))
print(list(filter(ok, range(1, 257)))) # Michael S. Branicky, Jun 22 2021
CROSSREFS
Column 1 of A240062.
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 02 2010
EXTENSIONS
Edited by N. J. A. Sloane, Sep 09 2023
Edited by Peter Munn, Oct 17 2023
STATUS
approved