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A129254 Numbers k such that both k and k+1 have at least one divisor of the form p^e with p<=e, p prime.

%I A129254 #13 Sep 24 2024 09:25:28
%S A129254 27,80,135,188,243,296,351,404,459,512,567,620,675,728,783,836,891,
%T A129254 944,999,1052,1107,1160,1215,1268,1323,1376,1431,1484,1539,1592,1647,
%U A129254 1700,1755,1808,1863,1916,1971,2024,2079,2132,2187,2240,2295,2348,2403,2456
%N A129254 Numbers k such that both k and k+1 have at least one divisor of the form p^e with p<=e, p prime.
%C A129254 From _Amiram Eldar_, Sep 23 2024: (Start)
%C A129254 This sequence is infinite: For example, if k is even then (2*k+1)*27 is a term, and if k is odd then (2*k+1)*27-1 is a term.
%C A129254 The numbers of terms that do not exceed 10^k, for k = 2, 3, ..., are 2, 19, 187, 1868, 18686, 186851, 1868507, 18685075, 186850742, ... . Apparently, the asymptotic density of this sequence exists and equals 0.01868507... . (End)
%H A129254 Amiram Eldar, <a href="/A129254/b129254.txt">Table of n, a(n) for n = 1..10000</a>
%F A129254 A129251(a(n)) > 0, A129251(a(n)+1) > 0.
%F A129254 If A100716(k) = a(n) then: A100716(k+1) = a(n) + 1.
%e A129254 135 = 5*3^3 and 135+1 = 136 = 17*2^3, therefore 135 is a term: a(3) = 135.
%e A129254 188 = 47*2^2 and 188+1 = 189 = 7*3^3, therefore 188 is a term: a(4) = 188.
%t A129254 SequencePosition[Table[If[AnyTrue[FactorInteger[n],#[[2]]>=#[[1]]&],1,0],{n,2500}],{1,1}][[All,1]] (* _Harvey P. Dale_, Sep 14 2019 *)
%o A129254 (PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,1] <= f[i,2], return(1))); 0;}
%o A129254 lista(kmax) = {my(is1 = 0, is2); for(k = 2, kmax, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);} \\ _Amiram Eldar_, Sep 23 2024
%Y A129254 Subsequence of A068781 and A100716.
%K A129254 nonn
%O A129254 1,1
%A A129254 _Reinhard Zumkeller_, Apr 07 2007