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 A003680 Smallest number with 2n divisors. (Formerly M1586) 22

%I M1586

%S 2,6,12,24,48,60,192,120,180,240,3072,360,12288,960,720,840,196608,

%T 1260,786432,1680,2880,15360,12582912,2520,6480,61440,6300,6720,

%U 805306368,5040,3221225472,7560,46080,983040,25920,10080,206158430208,3932160,184320,15120

%N Smallest number with 2n divisors.

%C Refers to the least number which is multiplicatively n-perfect, i.e. least number m the product of whose divisors equals m^n. - _Lekraj Beedassy_, Sep 18 2004

%C For n=1 to 5, a(n) equals second term of A008578, A007422, A162947, A048945, A030628. - _Michel Marcus_, Feb 04 2014

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 23.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vaclav Kotesovec, <a href="/A003680/b003680.txt">Table of n, a(n) for n = 1..3300</a> (terms 1..1000 from T. D. Noe using A005179)

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%F Bisection of A005179(n). - _Lekraj Beedassy_, Sep 21 2004

%t A005179 = Cases[Import["https://oeis.org/A005179/b005179.txt", "Table"], {_, _}][[All, 2]];

%t A = {#, DivisorSigma[0, #]}& /@ A005179;

%t a[n_] := SelectFirst[A, #[[2]] == 2n&][[1]];

%t a /@ Range[1000] (* _Jean-François Alcover_, Nov 10 2019 *)

%t mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[m < n, {}, {{n}}]; mp[n_, m_] := Join @@ Table[Map[Prepend[#, d] &, mp[n/d, d]], {d, Select[Rest[Divisors[n]], # <= m &]}]; mp[n_] := mp[n, n]; Table[mulpar = mp[2*n] - 1; Min[Table[Product[Prime[s]^mulpar[[j, s]], {s, 1, Length[mulpar[[j]]]}], {j, 1, Length[mulpar]}]], {n, 1, 100}] (* _Vaclav Kotesovec_, Apr 04 2021 *)

%o (PARI) a(n)=my(k=2*n); while(numdiv(k)!=2*n, k++); k \\ _Charles R Greathouse IV_, Jun 23 2017

%o (Python)

%o from sympy import divisors

%o def a(n):

%o m = 4*n - 2

%o while len(divisors(m)) != 2*n: m += 1

%o return m

%o print([a(n) for n in range(1, 19)]) # _Michael S. Branicky_, Feb 06 2021

%Y Cf. A005179 (n), A061283 (2n-1), A118224 (at least 2n).

%K nonn,easy,nice,changed

%O 1,1

%A _N. J. A. Sloane_, _Mira Bernstein_

%E More terms from _Jud McCranie_ Oct 15 1997

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Last modified April 17 11:45 EDT 2021. Contains 343064 sequences. (Running on oeis4.)