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A033950 Refactorable numbers: number of divisors of n divides n. Also known as tau numbers. 107
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 (list; graph; refs; listen; history; text; internal format)
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 n such that the equation GCD(n,x)=tau(n) 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)) - ceil(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(n) = n mod A000005(n). - 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

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, B12.

New Scientist, Sep 05 1998, p. 17, para. 3.

LINKS

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A. Bundy, Simon Colton, T. Walsh, HR - A system for Machine Discovery in Finite Algebras, ECAI 1998.

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.

S. 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, 13:2 (1990), pp. 383-386.

Claudia Spiro, How often is the number of divisors of n a divisor of n?, J. Number Theory 21 (1985), no. 1, 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

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 *)

PROG

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

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

(Haskell)

a033950 n = a033950_list !! (n-1)

a033950_list = [x | x <- [1..], x `mod` a000005 x == 0]

-- Reinhard Zumkeller, Dec 28 2011

CROSSREFS

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

Cf. A235353 (subsequence).

Cf. A054008.

Sequence in context: A086678 A066550 A162952 * A046526 A057529 A120737

Adjacent sequences:  A033947 A033948 A033949 * A033951 A033952 A033953

KEYWORD

nonn,nice

AUTHOR

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

EXTENSIONS

More terms from Erich Friedman

STATUS

approved

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Last modified December 8 04:27 EST 2016. Contains 278902 sequences.