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A033950
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Refactorable numbers: number of divisors of n divides n. Also known as tau numbers.
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81
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| Kennedy and Cooper show that this sequence has density zero.
Numbers n such that the equation gcd(n,x)=tau(n) has solutions - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 10 2002
Refactorable numbers are the fixed points of A009230. - Labos E. (labos(AT)ana.sote.hu), Nov 18 2002
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REFERENCES
| Colton, S., Bundy, A. and Walsh, T. R. S., HR - A system for machine discovery in finite algebras, ECAI 98, forthcoming.
R. K. Guy, Unsolved Problems in Number Theory, B12.
Kennedy, Robert E. and Cooper, Curtis N.; Tau numbers, natural density and Hardy and Wright's Theorem 437, International Journal of Mathematics and Mathematical Sciences, 13 (1990), no. 2, 383-386.
New Scientist, 5th Sept. 1998, p. 17, para. 3.
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LINKS
| Charles R Greathouse IV, Table of n, a(n) for n = 1..10000, replacing an earlier b-file from T. D. Noe
A. Bundy, Simon Colton, T. Walsh, HR- A system for Machine Discovery in Finite Algebras, ECAI 1998. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 29 2010]
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8
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MAPLE
| with(numtheory): A033950 := proc(n) option remember: local k: if(n=1)then return 1: else k:=procname(n-1)+1: do if(type(k/tau(k), integer))then return k: fi: k:=k+1: od: fi: end: seq(A033950(n), n=1..56); # Nathaniel Johnston, May 04 2011
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MATHEMATICA
| Do[If[IntegerQ[n/DivisorSigma[0, n]], Print[n]], {n, 1, 1000}]
Select[ Range[559], Mod[ #, DivisorSigma[0, # ]] == 0 &]
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PROG
| (MAGMA) [ n: n in [1..540] | n mod #Divisors(n) eq 0 ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), 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
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CROSSREFS
| Cf. A000005, A039819, A036762, A051278, A051279, A051280, A036763.
Sequence in context: A086678 A066550 A162952 * A046526 A057529 A120737
Adjacent sequences: A033947 A033948 A033949 * A033951 A033952 A033953
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KEYWORD
| nonn,nice
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AUTHOR
| Simon Colton (simonco(AT)cs.york.ac.uk)
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EXTENSIONS
| More terms from Erich Friedman (erich.friedman(AT)stetson.edu)
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