

A174973


Numbers whose divisors increase by a factor of 2 or less.


35



1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 256
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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 "2densely divisible". In general they say that n is ydensely 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 ysmooth.  David S. Metzler, Jul 02 2013
Also, numbers n with the property that the symmetric representation of sigma(n) has only one part. See A238443 and A237593.  Omar E. Pol, Mar 06 2014
Saias (1997) called these terms "2dense 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
Subsequence of A071562. All powers of 2 (cf. A000079) and all even perfect numbers (cf. A000396) are in the sequence.  Omar E. Pol, Aug 04 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 2densely Divisible Numbers, Integers, Vol. 20 (2020), Article A2.
Eric Saias, Entiers à diviseurs denses 1, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163191.
Terence Tao, A Truncated Elementary Selberg Sieve of Pintz. (blog entry defining ydensely 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. 743758; arXiv preprint, arXiv:1405.2585 [math.NT], 20142015.
Andreas Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp., Vol. 88, No. 318 (2019), pp. 18831902; arXiv preprint, arXiv:1705.06349 [math.NT], 20172018.


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.


MAPLE

a:= proc() option remember; local k; for k from 1+`if`(n=1, 0,
a(n1)) while (l> ormap(x> x, [seq(l[i]>l[i1]*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 !! (n1)
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[i1], 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[i1] <= 2 for i in range(1, len(d)))
print(list(filter(ok, range(1, 257)))) # Michael S. Branicky, Jun 22 2021


CROSSREFS

Subsequence of A196149. Same as A238443.
Cf. A027750, A047836, A237593.
Sequence in context: A125225 A092903 A005153 * A238443 A325795 A325781
Adjacent sequences: A174970 A174971 A174972 * A174974 A174975 A174976


KEYWORD

nonn


AUTHOR

T. D. Noe, Apr 02 2010


STATUS

approved



