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A358686 Numbers sandwiched between two semiprimes, one of which is a square.

%I A358686 #30 Jul 23 2023 01:53:43
%S A358686 5,50,120,122,288,290,528,842,960,1370,1680,1850,2808,2810,4488,5328,
%T A358686 5330,6240,6242,6888,6890,9408,9410,11880,12768,18770,22200,22800,
%U A358686 26568,27888,36482,38808,39600,52440,54290,58080,63000,63002,69170,72360,72362,73442,76730,78960
%N A358686 Numbers sandwiched between two semiprimes, one of which is a square.
%C A358686 Numbers in A124936 but not in A358665.
%C A358686 All numbers except 5 (the first term) are even.
%C A358686 Subsequence of A124936.
%H A358686 Harvey P. Dale, <a href="/A358686/b358686.txt">Table of n, a(n) for n = 1..292</a> (all terms up to 10 million)
%e A358686 5 is sandwiched between two semiprimes 4 = 2*2 and 6 = 3*2, one of which is a square. Thus, 5 is in this sequence.
%e A358686 34 is sandwiched between squarefree semiprimes 33 = 3*11 and 35 = 5*7. Thus, 34 is not in this sequence.
%t A358686 Select[Range[100000], Total[Transpose[FactorInteger[# - 1]][[2]]] == 2 && Total[Transpose[FactorInteger[# + 1]][[2]]] == 2 && ! (Transpose[FactorInteger[# - 1]][[2]] == {1, 1} && Transpose[FactorInteger[# + 1]][[2]] == {1, 1}) &]
%t A358686 Mean/@Select[SequencePosition[PrimeOmega[Range[80000]],{2,_,2}],AnyTrue[Sqrt[#],IntegerQ]&] (* _Harvey P. Dale_, Jun 14 2023 *)
%o A358686 (Python)
%o A358686 from sympy import factorint
%o A358686 from itertools import count, islice
%o A358686 def agen(): # generator of terms
%o A358686     nxt = []
%o A358686     yield 5
%o A358686     for k in count(6, 2):
%o A358686         prv, nxt = nxt, list(factorint(k+1).values())
%o A358686         if (prv==[1, 1] and nxt==[2]) or (prv==[2] and nxt==[1, 1]): yield k
%o A358686 print(list(islice(agen(), 44))) # _Michael S. Branicky_, Nov 26 2022
%Y A358686 Cf. A001358, A006881, A124936, A358665.
%K A358686 nonn
%O A358686 1,1
%A A358686 _Tanya Khovanova_, Nov 26 2022