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p = Prime[Range[PrimePi[10^3]]];
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SequenceCases[p, ({a_, b_, c_}) /; (a + c < 2 b) :> b, 36, Overlaps -> True]
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SequenceCases[p, ({a_, b_, c_}) /; (a + c > 2 b) :> b, 37, Overlaps -> True]
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p = Prime[Range[PrimePi[10^6] + 1]];
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Length[Select[Partition[p, 3, 1], #[[3]] + #[[1]] < 2 #[[2]] &]]
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Length[Select[Partition[p, 3, 1], #[[3]] + #[[1]] > 2 #[[2]] &]]
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p = Prime[Range[PrimePi[10^7] + 1]];
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Length[Select[Partition[p, 3, 1], #[[3]] + #[[1]] < 2 #[[2]] &]]
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Length[Select[Partition[p, 3, 1], #[[3]] + #[[1]] > 2 #[[2]] &]]
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