cp's OEIS Frontend

A003680 Smallest number with 2n divisors.

%I A003680 M1586 #42 Jul 06 2025 15:07:08
%S A003680 2,6,12,24,48,60,192,120,180,240,3072,360,12288,960,720,840,196608,
%T A003680 1260,786432,1680,2880,15360,12582912,2520,6480,61440,6300,6720,
%U A003680 805306368,5040,3221225472,7560,46080,983040,25920,10080,206158430208,3932160,184320,15120
%N A003680 Smallest number with 2n divisors.
%C A003680 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 A003680 For n=1 to 5, a(n) equals second term of A008578, A007422, A162947, A048945, A030628. - _Michel Marcus_, Feb 04 2014
%D A003680 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 A003680 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 23.
%D A003680 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003680 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 A003680 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 A003680 Bisection of A005179(n). - _Lekraj Beedassy_, Sep 21 2004
%t A003680 A005179 = Cases[Import["https://oeis.org/A005179/b005179.txt", "Table"], {_, _}][[All, 2]];
%t A003680 A = {#, DivisorSigma[0, #]}& /@ A005179;
%t A003680 a[n_] := SelectFirst[A, #[[2]] == 2n&][[1]];
%t A003680 a /@ Range[1000] (* _Jean-François Alcover_, Nov 10 2019 *)
%t A003680 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 *)
%t A003680 With[{tbl=Table[{n,DivisorSigma[0,n]},{n,800000}]},Table[SelectFirst[tbl,#[[2]]==2k&],{k,20}]][[;;,1]] (* The program generates the first 20 terms of the sequence. *) (* _Harvey P. Dale_, Jul 06 2025 *)
%o A003680 (PARI) a(n)=my(k=2*n); while(numdiv(k)!=2*n, k++); k \\ _Charles R Greathouse IV_, Jun 23 2017
%o A003680 (Python)
%o A003680 from sympy import divisors
%o A003680 def a(n):
%o A003680   m = 4*n - 2
%o A003680   while len(divisors(m)) != 2*n: m += 1
%o A003680   return m
%o A003680 print([a(n) for n in range(1, 19)]) # _Michael S. Branicky_, Feb 06 2021
%Y A003680 Cf. A005179 (n), A061283 (2n-1), A118224 (at least 2n).
%K A003680 nonn,easy,nice
%O A003680 1,1
%A A003680 _N. J. A. Sloane_, _Mira Bernstein_
%E A003680 More terms from _Jud McCranie_ Oct 15 1997