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A365867 Numbers k such that k and k+1 are both divisible by the cube of their least prime factor.

%I A365867 #8 Sep 21 2023 06:29:26
%S A365867 80,135,296,343,351,512,567,624,728,783,944,999,1160,1215,1375,1376,
%T A365867 1431,1592,1624,1647,1808,1863,2024,2079,2240,2295,2375,2400,2456,
%U A365867 2511,2672,2727,2888,2943,3104,3159,3320,3375,3536,3591,3624,3752,3807,3968,4023,4184
%N A365867 Numbers k such that k and k+1 are both divisible by the cube of their least prime factor.
%C A365867 Numbers k such that k and k+1 are both terms of A365866.
%C A365867 The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are , 12, 110, 1119, 11167, 111662, 1116693, 11166978, 111669826, 1116697990, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0111669... .
%H A365867 Amiram Eldar, <a href="/A365867/b365867.txt">Table of n, a(n) for n = 1..10000</a>
%e A365867 80 is a term since 2 is the least prime factor of 80 and 80 is divisible by 2^3 = 8, and 3 is the least prime factor of 81 and 81 is divisible by 3^3 = 27.
%t A365867 q[n_] := FactorInteger[n][[1, -1]] >= 3; 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[5000]
%o A365867 (PARI) lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = factor(k)[1,2] >= 3; if(q1 && q2, print1(k-1, ", ")); q1 = q2);}
%Y A365867 Cf. A067029.
%Y A365867 Subsequence of A068140 and A365866.
%Y A365867 A365868 is a subsequence.
%K A365867 nonn,easy
%O A365867 1,1
%A A365867 _Amiram Eldar_, Sep 21 2023