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A361209 Second hexagonal numbers having middle divisors.

%I A361209 #35 Oct 23 2023 08:34:43
%S A361209 36,210,300,528,990,1176,1485,1596,2080,2346,3240,3570,4095,4278,4851,
%T A361209 5460,6555,6786,7260,8256,8778,9870,10440,11628,12880,13530,14196,
%U A361209 14535,15225,15576,17020,17766,20100,20910,21736,22578,23436,24310,25200,26565,27495,27966,30876
%N A361209 Second hexagonal numbers having middle divisors.
%C A361209 The middle divisors of n are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).
%C A361209 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.
%H A361209 Paolo Xausa, <a href="/A361209/b361209.txt">Table of n, a(n) for n = 1..10000</a>
%e A361209 36 is in the sequence because it is a second hexagonal number (A014105) and it has a middle divisor, the 6.
%e A361209 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.
%t A361209 A071562Q[n_]:=With[{m1=Sqrt[n/2],m2=Sqrt[2n]},DivisorSum[n,#&,m1<=#<m2&]>0];
%t A361209 With[{upto=200},Select[Array[#(2#+1)&,upto],A071562Q]] (* Checks the first 200 second hexagonal numbers *) (* _Paolo Xausa_, Oct 23 2023 *)
%o A361209 (PARI) hasmd(n)=fordiv(n, d, if(d^2>=n/2 && d^2<2*n, return(1))); 0; \\ A014105
%o A361209 select(hasmd, vector(150, n, n*(2*n + 1))) \\ _Michel Marcus_, Mar 10 2023
%Y A361209 Intersection of A014105 and A071562.
%Y A361209 Nonzero terms of A014107 without the terms of A298856.
%Y A361209 Cf. A067742, A236104, A237048, A237591, A237593, A262626, A240542.
%K A361209 nonn
%O A361209 1,1
%A A361209 _Omar E. Pol_, Mar 10 2023
%E A361209 More terms from _Michel Marcus_, Mar 10 2023