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A365890 Numbers k such that k and k+1 are both terms of A365889.

%I A365890 #11 Sep 24 2023 09:16:54
%S A365890 27,188,459,620,675,783,836,944,1107,1268,1323,1484,1647,1755,1808,
%T A365890 1916,1971,2132,2240,2403,2564,2619,2780,3051,3124,3212,3267,3375,
%U A365890 3428,3536,3644,3699,3860,3915,4076,4239,4347,4400,4508,4563,4671,4724,4995,5103,5156
%N A365890 Numbers k such that k and k+1 are both terms of A365889.
%C A365890 The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 8, 88, 862, 8607, 86044, 860407, 8604097, 86041005, 860410068, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00860410... .
%H A365890 Amiram Eldar, <a href="/A365890/b365890.txt">Table of n, a(n) for n = 1..10000</a>
%e A365890 27 = 3^3 is a term since its least prime factor, 3, divides its exponent, 3, and the least prime factor of 28 = 2^2 * 7, 2, also divides its exponent, 2.
%e A365890 783 = 3^3 * 29 is a term since its least prime factor, 3, divides its exponent, 3, and the least prime factor of 784 = 2^4 * 7^2, 2, also divides its exponent, 4.
%t A365890 q[n_] := Divisible @@ Reverse[FactorInteger[n][[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[6000]
%o A365890 (PARI) is(n) = {my(f = factor(n)); n > 1 && !(f[1, 2] % f[1, 1]);}
%o A365890 lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = is(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}
%Y A365890 Subsequence of A365889.
%Y A365890 Subsequences: A365884, A365891.
%K A365890 nonn,easy
%O A365890 1,1
%A A365890 _Amiram Eldar_, Sep 22 2023