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A365864 Numbers k such that k and k+1 are both divisible by the square of their least prime factor.

%I A365864 #10 Sep 21 2023 06:28:59
%S A365864 8,24,27,44,48,63,80,99,116,120,124,135,152,168,171,175,188,207,224,
%T A365864 243,260,275,279,288,296,315,324,332,343,351,360,368,387,404,423,424,
%U A365864 440,459,475,476,495,512,528,531,539,548,567,575,584,603,620,624,636,639
%N A365864 Numbers k such that k and k+1 are both divisible by the square of their least prime factor.
%C A365864 Numbers k such that k and k+1 are both terms of A283050.
%C A365864 The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 8, 82, 802, 8009, 80078, 800900, 8009533, 80097354, 800979764, 8009809838, ... . Apparently, the asymptotic density of this sequence exists and equals 0.08009... .
%H A365864 Amiram Eldar, <a href="/A365864/b365864.txt">Table of n, a(n) for n = 1..10000</a>
%e A365864 8 is a term since 2 is the least prime factor of 8 and 8 is divisible by 2^2 = 4, and 3 is the least prime factor of 9 and 9 is divisible by 3^3 = 9.
%t A365864 q[n_] := FactorInteger[n][[1, -1]] >= 2; 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[640]
%o A365864 (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 A365864 Cf. A067029.
%Y A365864 Subsequence of A068781 and A283050.
%Y A365864 A365865 is a subsequence.
%K A365864 nonn,easy
%O A365864 1,1
%A A365864 _Amiram Eldar_, Sep 21 2023