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A348079 Starts of runs of 5 consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039).

%I A348079 #7 Sep 28 2021 04:50:28
%S A348079 792007675,2513546971,2820448771,3201296272,4742326672,4894282924,
%T A348079 5462510272,5664816448,6947006272,7814337424,8784450448,9085360624,
%U A348079 10147712524,10246365547,11537724975,11861786572,11907710548,12456672496,13338112048,13510075471,13931933948
%N A348079 Starts of runs of 5 consecutive numbers that have an equal number of even and odd exponents in their prime factorization (A187039).
%e A348079 792007675 is a term since 792007675 = 2^2 * 31680307, 792007675 + 1 = 792007676 = 2^2 * 198001919, 792007675 + 2 = 792007677 = 3^2 * 88000853, 792007675 + 3 = 792007678 = 2 * 7^2 * 11^2 * 66791 and 792007675 + 4 = 792007679 = 17^2 * 2740511 all have an equal number of even and odd exponents in their prime factorization.
%t A348079 q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), _?OddQ] == Count[e, _?EvenQ]; v = q /@ Range[5]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 4]], {k, 6, 3*10^9}]; seq
%o A348079 (Python)
%o A348079 from sympy import factorint
%o A348079 def cond(n):
%o A348079     evenodd = [0, 0]
%o A348079     for e in factorint(n).values():
%o A348079         evenodd[e%2] += 1
%o A348079     return evenodd[0] == evenodd[1]
%o A348079 def afind(limit, startk=6):
%o A348079     condvec = [cond(startk+i) for i in range(5)]
%o A348079     for kp4 in range(startk+4, limit+5):
%o A348079         condvec = condvec[1:] + [cond(kp4)]
%o A348079         if all(condvec):
%o A348079             print(kp4-4, end=", ")
%o A348079 afind(10**9) # _Michael S. Branicky_, Sep 27 2021
%Y A348079 Subsequence of A187039, A348076, A348077 and A348078.
%K A348079 nonn
%O A348079 1,1
%A A348079 _Amiram Eldar_, Sep 27 2021