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%I #22 Nov 01 2020 21:42:48
%S 900,1764,1800,2700,3528,3600,4356,4500,4900,5292,5400,6084,6300,7056,
%T 7200,8100,8712,8820,9000,9800,9900,10404,10584,10800,11025,11700,
%U 12100,12168,12348,12600,12996,13068,13500,14112,14400,14700,15300,15876,16200,16900
%N Numbers k such that there exists a strict relatively prime factorization of k in which no pair of factors is relatively prime.
%C From _Amiram Eldar_, Nov 01 2020: (Start)
%C Also, numbers with more than two non-unitary prime divisors, i.e., numbers k such that A056170(k) > 2, or equivalently, numbers divisible by the squares of three distinct primes.
%C The complement of the union of A005117, A190641 and A338539.
%C The asymptotic density of this sequence is 1 - 6/Pi^2 - (6/Pi^2)*A154945 - (3/Pi^2)*(A154945^2 - A324833) = 0.0033907041... (End)
%H Amiram Eldar, <a href="/A318720/b318720.txt">Table of n, a(n) for n = 1..10000</a>
%e 900 is in the sequence because the factorization 900 = (6*10*15) is relatively prime (since the GCD of (6,10,15) is 1) but each of the pairs (6,10), (6,15), (10,15) has a common divisor > 1. Larger examples are:
%e 1800 = (6*15*20) = (10*12*15).
%e 9900 = (6*10*165) = (6*15*110) = (10*15*66).
%e 5400 = (6*20*45) = (10*12*45) = (10*15*36) = (15*18*20).
%e 60 is not in the sequence because all its possible factorizations (4 * 15, 3 * 4 * 5, etc.) contain at least one pair that is coprime, if not more than one prime.
%t strfacs[n_] := If[n <= 1, {{}}, Join@@Table[(Prepend[#1, d] &)/@Select[strfacs[n/d], Min@@#1 > d &], {d, Rest[Divisors[n]]}]]; Select[Range[10000], Function[n, Select[strfacs[n], And[GCD@@# == 1, And@@(GCD[##] > 1 &)@@@Select[Tuples[#, 2], Less@@# &]] &] != {}]]
%t Select[Range[20000], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] > 2 &] (* _Amiram Eldar_, Nov 01 2020 *)
%Y Cf. A001055, A001221, A001222, A007716, A045778, A051185, A078374, A281116, A303140, A303283, A305843, A305854, A317748, A318715, A318717, A318721.
%Y Cf. A005117, A036785, A056170, A154945, A190641, A324833, A338539.
%K nonn
%O 1,1
%A _Gus Wiseman_, Sep 02 2018
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