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A365865 Starts of runs of 3 consecutive integers that are divisible by the square of their least prime factor.

%I A365865 #11 Aug 05 2024 19:43:40
%S A365865 423,475,1323,1375,1519,2007,2223,2275,2871,3123,3175,3211,3283,3479,
%T A365865 3575,3751,3771,4023,4075,4475,4923,4959,4975,5047,5535,5823,5875,
%U A365865 6723,6775,6811,7299,7623,7675,8107,8379,8523,8575,8955,9423,9475,10323,10339,10375,10467
%N A365865 Starts of runs of 3 consecutive integers that are divisible by the square of their least prime factor.
%C A365865 Numbers k such that k, k+1 and k+2 are all terms of A283050.
%C A365865 Numbers of the form 4*k+2 are not terms of A283050. Therefore, there are no runs of 4 or more consecutive integers, and all the terms of this sequence are of the form 4*k+3.
%C A365865 The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 2, 40, 429, 4419, 44352, 444053, 4441769, 44421000, 444220814, ... . Apparently, the asymptotic density of this sequence exists and equals 0.004442... .
%H A365865 Amiram Eldar, <a href="/A365865/b365865.txt">Table of n, a(n) for n = 1..10000</a>
%e A365865 423 is a term since 3 is the least prime factor of 423 and 423 is divisible by 3^2 = 9, 2 is the least prime factor of 424 and 424 is divisible by 2^2 = 4, and 5 is the least prime factor of 425 and 425 is divisible by 5^2 = 25.
%t A365865 Select[4 * Range[2700] + 3, AllTrue[# + {0, 1, 2}, FactorInteger[#1][[1, -1]] >= 2 &] &]
%t A365865 SequencePosition[Table[If[Divisible[n,FactorInteger[n][[1,1]]^2],1,0],{n,11000}],{1,1,1}][[;;,1]] (* _Harvey P. Dale_, Aug 05 2024 *)
%o A365865 (PARI) is(n) = factor(n)[1,2] >= 2;
%o A365865 lista(kmax) = forstep(k = 3, kmax, 4, if(is(k) && is(k+1) && is(k+2), print1(k, ", ")));
%Y A365865 Cf. A067029.
%Y A365865 Subsequence of A004767, A070258, A283050 and A365864.
%K A365865 nonn,easy
%O A365865 1,1
%A A365865 _Amiram Eldar_, Sep 21 2023