A033950 Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.

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

Offset: 1

Author

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

Keywords

• nonn
• nice

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

Refactorable numbers are the fixed points of A009230. - Labos Elemer, Nov 18 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

A054008(k) = k mod A000005(k). - Reinhard Zumkeller, Sep 17 2014

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 k occurs as tau(m) for some m in the sequence. If the factorization of k is Product p_i^e_i, then Product p_i^(p_i^e_i-1) has the specified property. For k prime, this is the only such number. - Franklin T. Adams-Watters, 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

Numbers m such that the ratio (number of non-divisors of m)/(number of divisors of m) = A049820(m)/A000005(m) is an integer. - Michel Lagneau, Apr 04 2025

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, pp. 102-103.
• New Scientist, Sep 05 1998, p. 17, para. 3.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Kushagr Ahuja, Patrick Lei and Dylan Pentland, Tau ideals in number fields, PROMYS 2017.
• Alan Bundy, Simon Colton and Toby Walsh, HR - A system for Machine Discovery in Finite Algebras, ECAI 1998.
• Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
• Simon Colton, HR - Automatic Theory Formation in Pure Mathematics.
• Robert E. Kennedy and Curtis N. Cooper, Tau numbers, natural density and Hardy and Wright's Theorem 437, International Journal of Mathematics and Mathematical Sciences, Vol. 13, No. 2 (1990), pp. 383-386.
• Claudia Spiro, How often is the number of divisors of n a divisor of n?, J. Number Theory, Vol. 21, No. 1 (1985), pp. 81-100.
• Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Crossrefs

Cf. A000005, A000720, A039819, A036762, A051278, A051279, A051280, A036763.

Cf. A235353 (subsequence).

Cf. A054008, A281188.

Programs

Haskell

a033950 n = a033950_list !! (n-1)
a033950_list = [x | x <- [1..], x `mod` a000005 x == 0]
-- Reinhard Zumkeller, Dec 28 2011

Magma

[ n: n in [1..540] | n mod #Divisors(n) eq 0 ]; // Klaus Brockhaus, Apr 29 2009

Maple

with(numtheory):
A033950 := proc(n)
    option remember:
    local k:
    if n=1 then
        return 1:
    else
        for k from procname(n-1)+1 do
            if type(k/tau(k), integer) then
                return k:
            end if:
        end do:
    end if:
end proc:
seq(A033950(n), n=1..56); # Nathaniel Johnston, May 04 2011

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 *)

PARI

isA033950(n)=n%numdiv(n)==0 \\ Charles R Greathouse IV, Jun 10 2011

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

Extensions

More terms from Erich Friedman