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

0,1

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

Table of n, a(n) for n=0..36.

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: A363196 A025230 A152456 * A051511 A272030 A340878

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