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A365870 Numbers k such that k and k+1 both have an even exponent of least prime factor in their prime factorization.

%I A365870 #12 May 08 2024 08:07:58
%S A365870 44,48,63,80,99,116,171,175,207,260,275,315,324,332,368,387,404,475,
%T A365870 476,495,528,531,539,548,575,603,624,636,656,692,724,747,764,819,832,
%U A365870 891,908,924,931,960,963,980,1024,1035,1052,1071,1075,1088,1124,1179,1196,1232
%N A365870 Numbers k such that k and k+1 both have an even exponent of least prime factor in their prime factorization.
%C A365870 Numbers k such that k and k+1 are both terms of A365869.
%C A365870 The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 5, 42, 414, 4173, 41927, 419597, 4196917, 41972747, 419738185, 4197406018, ... . Apparently, the asymptotic density of this sequence exists and equals 0.04197... .
%H A365870 Amiram Eldar, <a href="/A365870/b365870.txt">Table of n, a(n) for n = 1..10000</a>
%e A365870 44 is a term since the exponent of the prime factor 2 in the factorization 44 = 2^2 * 11 is 2, which is even, and the exponent of the prime factor 3 in the factorization 45 = 3^2 * 5 is also 2, which is even.
%t A365870 q[n_] := EvenQ[FactorInteger[n][[1, -1]]]; consec[kmax_] := Module[{m = 1, c = Table[False, {2}], s = {}}, Do[c = Join[Rest[c], {q[k]}]; If[And @@ c, AppendTo[s, k - 1]], {k, 1, kmax}]; s]; consec[1250]
%o A365870 (PARI) lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = !(factor(k)[1,2]%2); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}
%Y A365870 Subsequence of A365864 and A365869.
%Y A365870 A365871 is a subsequence.
%K A365870 nonn,easy
%O A365870 1,1
%A A365870 _Amiram Eldar_, Sep 21 2023