%I #9 Sep 21 2023 06:29:04

%S 475,1519,2223,2275,3283,4475,4923,4975,5823,6723,6811,7299,7675,8107,

%T 8379,8523,8955,9475,10323,10467,11275,12427,12463,12591,13075,13867,

%U 13923,14355,15631,15723,16675,18027,18275,18475,18767,19323,19375,19647,22075,22831

%N Starts of runs of 3 consecutive integers whose exponent of least prime factor in their prime factorization is even.

%C Numbers k such that k, k+1 and k+2 are all terms of A365869.

%C Numbers of the form 4*k+2 are not terms of A365869. 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 The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 1, 18, 195, 1952, 19542, 195514, 1955859, 19560453, 195611458, ... . Apparently, the asymptotic density of this sequence exists and equals 0.001956... .

%H Amiram Eldar, <a href="/A365871/b365871.txt">Table of n, a(n) for n = 1..10000</a>

%e 475 is a term since the exponent of the prime factor 5 in the factorization 475 = 5^2 * 19 is 2, which is even, the exponent of the prime factor 2 in the factorization 476 = 2^2 * 7 * 17 is 2, which is even, and the exponent of the prime factor 3 in the factorization 477 = 3^2 * 53 is also 2, which is even.

%t Select[4 * Range[6000] + 3, AllTrue[# + {0, 1, 2}, EvenQ[FactorInteger[#1][[1, -1]]] &] &]

%o (PARI) is(n) = !(factor(n)[1,2]%2);

%o lista(kmax) = forstep(k = 3, kmax, 4, if(is(k) && is(k+1) && is(k+2), print1(k, ", ")));

%Y Cf. A067029.

%Y Subsequence of A004767, A365869 and A365870.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Sep 21 2023