A003680 - OEIS

A003680

Smallest number with 2n divisors.
(Formerly M1586)

2, 6, 12, 24, 48, 60, 192, 120, 180, 240, 3072, 360, 12288, 960, 720, 840, 196608, 1260, 786432, 1680, 2880, 15360, 12582912, 2520, 6480, 61440, 6300, 6720, 805306368, 5040, 3221225472, 7560, 46080, 983040, 25920, 10080, 206158430208, 3932160, 184320, 15120

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OFFSET

1,1

COMMENTS

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

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

REFERENCES

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.

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

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

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..3300 (terms 1..1000 from T. D. Noe using A005179)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

FORMULA

Bisection of A005179(n). - Lekraj Beedassy, Sep 21 2004

MATHEMATICA

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

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

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

a /@ Range[1000] (* Jean-François Alcover, Nov 10 2019 *)

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 *)

PROG

(PARI) a(n)=my(k=2*n); while(numdiv(k)!=2*n, k++); k \\ Charles R Greathouse IV, Jun 23 2017

(Python)

from sympy import divisors

def a(n):

m = 4*n - 2

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

return m

print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Feb 06 2021

CROSSREFS

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

Sequence in context: A305104 A118224 A227068 * A337257 A051487 A111286

Adjacent sequences: A003677 A003678 A003679 * A003681 A003682 A003683

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Mira Bernstein

EXTENSIONS

More terms from Jud McCranie Oct 15 1997

STATUS

approved