

A033950


Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.


194



1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, 328, 344, 348, 360, 372, 376, 384, 396, 424, 441, 444, 448, 450, 468, 472, 480, 488, 492, 504, 516, 536
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OFFSET

1,2


COMMENTS

Kennedy and Cooper show that this sequence has density zero.
Spiro showed more precisely that the number of refactorable numbers less than x is asymptotic to (x/sqrt(log x))(log(log x))^(1+o(1)).  David Eppstein, Aug 25 2014
Numbers k such that the equation gcd(k,x) = tau(k) has solutions.  Benoit Cloitre, Jun 10 2002
Let ref(n) denote the characteristic function of the refactorable numbers. Then ref(n) = 1 + floor(n/d(n))  ceiling(n/d(n)), where d(n) is the number of divisors of n.  Wesley Ivan Hurt, Jan 09 2013, Feb 15 2013
An odd number with an even number of divisors cannot be in the sequence by definition. Therefore all odd terms are squares (A000290).  Ivan N. Ianakiev, Aug 25 2013
The only squarefree terms are 1 and 2: if x is a squarefree number that is a product of n distinct primes, its number of divisors is 2^n, so x is refactorable if it contains 2^n as a factor, but that makes it nonsquarefree unless n = 0, 1, hence x = 1, 2.  Waldemar Puszkarz, Jun 10 2016
Every positive integer occurs as tau(k) for some k in the sequence. If the factorization of k is Product p_i^e_i, then Product p_i^(p_i^e_i1) has the specified property. For k prime, this is the only such number.  Franklin T. AdamsWatters, Jan 14 2017
Zelinsky (2002) proved that for any j > 0 and for sufficiently large m the number of terms not exceeding m is > j*pi(m), where pi(m) = A000720(m).  Amiram Eldar, Feb 20 2021


REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, pp. 102103.
New Scientist, Sep 05 1998, p. 17, para. 3.


LINKS



MAPLE

with(numtheory):
option remember:
local k:
if n=1 then
return 1:
else
for k from procname(n1)+1 do
if type(k/tau(k), integer) then
return k:
end if:
end do:
end if:
end proc:


MATHEMATICA

Do[If[IntegerQ[n/DivisorSigma[0, n]], Print[n]], {n, 1, 1000}]
Select[ Range[559], Mod[ #, DivisorSigma[0, # ]] == 0 &]
Select[Range[550], Divisible[ #, DivisorSigma[0, # ]]&] (* Waldemar Puszkarz, Jun 10 2016 *)


PROG

(Magma) [ n: n in [1..540]  n mod #Divisors(n) eq 0 ]; // Klaus Brockhaus, Apr 29 2009
(Haskell)
a033950 n = a033950_list !! (n1)
a033950_list = [x  x < [1..], x `mod` a000005 x == 0]
(Python)
from sympy import divisor_count
print([n for n in range(1, 1001) if not n % divisor_count(n)]) # Indranil Ghosh, May 03 2017


CROSSREFS



KEYWORD

nonn,nice


AUTHOR

Simon Colton (simonco(AT)cs.york.ac.uk)


EXTENSIONS



STATUS

approved



