A361209 Second hexagonal numbers having middle divisors.

36, 210, 300, 528, 990, 1176, 1485, 1596, 2080, 2346, 3240, 3570, 4095, 4278, 4851, 5460, 6555, 6786, 7260, 8256, 8778, 9870, 10440, 11628, 12880, 13530, 14196, 14535, 15225, 15576, 17020, 17766, 20100, 20910, 21736, 22578, 23436, 24310, 25200, 26565, 27495, 27966, 30876

Offset: 1

Author

Omar E. Pol, Mar 10 2023

Keywords

• nonn

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.

Examples

			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.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A014105 and A071562.

Nonzero terms of A014107 without the terms of A298856.

Cf. A067742, A236104, A237048, A237591, A237593, A262626, A240542.

Programs

Mathematica

A071562Q[n_]:=With[{m1=Sqrt[n/2],m2=Sqrt[2n]},DivisorSum[n,#&,m1<=#0];
With[{upto=200},Select[Array[#(2#+1)&,upto],A071562Q]] (* Checks the first 200 second hexagonal numbers *) (* Paolo Xausa, Oct 23 2023 *)

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

Extensions

More terms from Michel Marcus, Mar 10 2023