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Starts of runs of 3 consecutive integers in which each member of the run has at least one divisor of the form p^e with p <= e, where p is a prime.
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%I #10 Sep 24 2024 09:29:05

%S 71874,109375,156248,181250,228123,265624,409374,446875,493748,518750,

%T 565623,603124,746874,784375,831248,856250,903123,940624,1084374,

%U 1121875,1168748,1193750,1240623,1278124,1421874,1459375,1506248,1531250,1578123,1615624,1759374,1796875

%N Starts of runs of 3 consecutive integers in which each member of the run has at least one divisor of the form p^e with p <= e, where p is a prime.

%C The start of the least run of 4 (and also 5) consecutive integers with this property is 3988418748.

%C The numbers of terms that do not exceed 10^k, for k = 5, 6, ..., are 1, 18, 178, 1783, 17845, 178458, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00001784... .

%H Amiram Eldar, <a href="/A376469/b376469.txt">Table of n, a(n) for n = 1..10000</a>

%e 71874 = 2 * 3^3 * 11^3 is a term since it is divisible by 3^3, 71875 = 5^5 * 23 is divisible by 5^5, and 71876 = 2^2 * 7 * 17 * 151 is divisible by 2^2.

%t q[n_] := q[n] = AnyTrue[FactorInteger[n], First[#] <= Last[#] &]; Select[Range[2*10^6], q[#] && q[#+1] && q[#+2] &]

%o (PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,1] <= f[i,2], return(1))); 0;}

%o lista(kmax) = {my(is1 = 0, is2 = 0, is3); for(k = 3, kmax, is3 = is(k); if(is1 && is2 && is3, print1(k-2, ", ")); is1 = is2; is2 = is3);}

%Y Subsequence of A100716, A070258 and A129254.

%K nonn

%O 1,1

%A _Amiram Eldar_, Sep 23 2024