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 A128605 Smallest number m having exactly n divisors d with sqrt(m/2) <= d < sqrt(2*m). 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS A067742(a(n)) = n and A067742(m) <> n for m < a(n). From Hartmut F. W. Hoft, Feb 06 2017: (Start) 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. 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) The first 37 terms are 13-smooth (see A080197). - David A. Corneth, Apr 07 2018 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. EXAMPLE 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 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 *) PROG (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 CROSSREFS Cf. A067742. Related to Dyck paths: A237593, A240542, A279286. Sequence in context: A221852 A025230 A152456 * A051511 A272030 A026499 Adjacent sequences:  A128602 A128603 A128604 * A128606 A128607 A128608 KEYWORD nonn AUTHOR Reinhard Zumkeller, Mar 14 2007 EXTENSIONS a(33)-a(37) from Hartmut F. W. Hoft, Feb 06 2017 STATUS approved

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Last modified August 21 05:32 EDT 2019. Contains 326162 sequences. (Running on oeis4.)