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%I #42 Apr 27 2019 20:24:03
%S 3,1,6,72,120,1800,840,3600,2520,28800,10080,88200,27720,259200,50400,
%T 176400,83160,352800,138600,3484800,277200,1411200,360360,2822400,
%U 831600,3175200,720720,6350400,1663200,31363200,1441440,28576800,2162160,12700800,3326400,21344400,4324320
%N Smallest number m having exactly n divisors d with sqrt(m/2) <= d < sqrt(2*m).
%C A067742(a(n)) = n and A067742(m) <> n for m < a(n).
%C From _Hartmut F. W. Hoft_, Feb 06 2017: (Start)
%C a(66)=86486400 has the largest index n with a(n) <= 100000000, but there are 12 values from a(38) to a(67) that are larger than 100000000.
%C Conjecture: a(n) = k where p(k) and p(k-1) are the first pair of Dyck paths for the symmetric representation of sigma(k) and sigma(k-1), as described in A237593, having a gap of exactly n units on the diagonal, i.e., it is the sequence of record gaps in sequence A240542; tested through 2000000 with a variant of function A279286. (End)
%C The first 37 terms are 13-smooth (see A080197). - _David A. Corneth_, Apr 07 2018
%H David A. Corneth, <a href="/A128605/a128605_4.gp.txt">Upper bounds on a(0)..a(376) and some more values</a>
%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1505.07229v3">The zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, 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.]
%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1610.07793 [math.NT], 2016, see Remark 1.3.
%e A067742(a(5)) = A067742(1800) = #{30,36,40,45,50} = 5;
%e A067742(a(6)) = A067742(840) = #{21,24,28,30,35,40} = 6;
%e A067742(a(7)) = A067742(3600) = #{45,48,50,60,72,75,80} = 7.
%e a(0)=3 since 3 has no middle divisors. - _Hartmut F. W. Hoft_, Feb 06 2017
%t (* computation based on the function of _Michael Somos_ in A067742 *)
%t 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]]
%t a128605[70,100000000] (* computes those a(0) .. a(66) <= 100000000 *)
%t (* _Hartmut F. W. Hoft_, Feb 06 2017 *)
%o (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)
%o 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
%Y Cf. A067742.
%Y Related to Dyck paths: A237593, A240542, A279286.
%K nonn
%O 0,1
%A _Reinhard Zumkeller_, Mar 14 2007
%E a(33)-a(37) from _Hartmut F. W. Hoft_, Feb 06 2017
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