%I
%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 n such that there exists a strict relatively prime factorization of n in which no pair of factors is relatively prime.
%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@@# &]] &] != {}]]
%Y Cf. A001055, A001221, A001222, A007716, A045778, A051185, A078374, A281116, A303140, A303283, A305843, A305854, A317748, A318715, A318717, A318721.
%K nonn
%O 1,1
%A _Gus Wiseman_, Sep 02 2018
