OFFSET
1,1
COMMENTS
The middle divisors of n are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).
Also numbers k with the property that in the symmetric representation of sigma(k) the smallest Dyck path has a central peak and the largest Dyck path has a central valley and both Dyck paths do not meet in the center.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..10000
EXAMPLE
36 is in the sequence because it is a second hexagonal number (A014105) and it has a middle divisor, the 6.
On the other hand the 35th row of A237593 is [18,7,3,2,2,1,2,2,1,2,2,3,7,18] and the 36th row of the same triangle is [19,6,4,2,2,1,1,1,1,1,1,2,2,4,6,19]. Since the smallest Dyck path of the symmetric representation of sigma(36) has a central peak and the largest Dyck path has a central valley and both Dyck paths do not meet in the center so 36 is in the sequence. The diagram is too large to include.
MATHEMATICA
A071562Q[n_]:=With[{m1=Sqrt[n/2], m2=Sqrt[2n]}, DivisorSum[n, #&, m1<=#<m2&]>0];
With[{upto=200}, Select[Array[#(2#+1)&, upto], A071562Q]] (* Checks the first 200 second hexagonal numbers *) (* Paolo Xausa, Oct 23 2023 *)
PROG
(PARI) hasmd(n)=fordiv(n, d, if(d^2>=n/2 && d^2<2*n, return(1))); 0; \\ A014105
select(hasmd, vector(150, n, n*(2*n + 1))) \\ Michel Marcus, Mar 10 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Mar 10 2023
EXTENSIONS
More terms from Michel Marcus, Mar 10 2023
STATUS
approved