A128605 Smallest number m having exactly n divisors d with sqrt(m/2) <= d < sqrt(2*m).
3, 1, 6, 72, 120, 1800, 840, 3600, 2520, 28800, 10080, 88200, 27720, 259200, 50400, 176400, 83160, 352800, 138600, 3484800, 277200, 1411200, 360360, 2822400, 831600, 3175200, 720720, 6350400, 1663200, 31363200, 1441440, 28576800, 2162160, 12700800, 3326400, 21344400, 4324320
Offset: 0
Keywords
Examples
A067742(a(5)) = A067742(1800) = #{30,36,40,45,50} = 5;
A067742(a(6)) = A067742(840) = #{21,24,28,30,35,40} = 6;
A067742(a(7)) = A067742(3600) = #{45,48,50,60,72,75,80} = 7.
a(0)=3 since 3 has no middle divisors. - _Hartmut F. W. Hoft_, Feb 06 2017
Links
- David A. Corneth, Upper bounds on a(0)..a(376) and some more values
- Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015, see page 29 Remarks 6.8(b). [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
- Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016, see Remark 1.3.
Programs
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Mathematica
(* computation based on the function of Michael Somos in A067742 *) a128605[pL_,b_] := Module[{posL=Map[0&, Range[pL]], k=1, mCur, count}, While[k<=b, mCur=DivisorSum[k, 1&, k/2 <= #^2 < 2k&]; If[posL[[mCur]]==0, posL[[mCur]]=k]; k++]; Prepend[posL, 3]] a128605[70,100000000] (* computes those a(0) .. a(66) <= 100000000 *) (* Hartmut F. W. Hoft, Feb 06 2017 *)
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PARI
ct(m)=my(lower=if(m%2==0&&issquare(m/2), sqrtint(m/2), sqrtint(m\2)+1), upper=sqrtint(2*m)); sumdiv(m, d, lower<=d && d<=upper) v=vector(10^3); need=1; for(m=1, 1e9, t=ct(m); if(t>=need && v[t]==0, v[t]=m; print("a("t") = "n); while(v[need], need++))) \\ Charles R Greathouse IV, Feb 06 2017
Extensions
a(33)-a(37) from Hartmut F. W. Hoft, Feb 06 2017
Comments