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A361209
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Second hexagonal numbers having middle divisors.
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1
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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
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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