A361209 - OEIS

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

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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

Intersection of A014105 and A071562.

Nonzero terms of A014107 without the terms of A298856.

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

Sequence in context: A339981 A304378 A120465 * A144523 A232677 A232670

Adjacent sequences: A361206 A361207 A361208 * A361210 A361211 A361212