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A316268
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FDH numbers of connected strict integer partitions.
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0
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2, 3, 4, 5, 7, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 36, 37, 39, 41, 43, 45, 47, 49, 51, 53, 59, 61, 64, 65, 67, 69, 71, 73, 79, 81, 83, 85, 87, 89, 92, 97, 101, 103, 107, 108, 109, 111, 113, 115, 117, 119, 121, 124, 127, 129, 131, 135, 137, 139, 144
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OFFSET
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1,1
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COMMENTS
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Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k).
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set or strict partition S is said to be connected if G(S) is a connected graph.
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LINKS
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EXAMPLE
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Sequence of connected strict integer partitions begins (1), (2), (3), (4), (5), (6), (7), (8), (4,2), (9), (10), (11), (12), (13), (6,2).
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MATHEMATICA
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nn=100;
FDfactor[n_]:=If[n===1, {}, Sort[Join@@Cases[FactorInteger[n], {p_, k_}:>Power[p, Cases[Position[IntegerDigits[k, 2]//Reverse, 1], {m_}->2^(m-1)]]]]];
FDrules=MapIndexed[(#1->#2[[1]])&, Array[FDfactor, nn, 1, Union]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>1]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[nn], Length[csm[primeMS/@(FDfactor[#]/.FDrules)]]==1&]
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CROSSREFS
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Cf. A048143, A050376, A064547, A213925, A299755, A299756, A299757, A304714, A304716, A305078, A305079, A305829, A305831.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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