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A318720
Numbers k such that there exists a strict relatively prime factorization of k in which no pair of factors is relatively prime.
8
900, 1764, 1800, 2700, 3528, 3600, 4356, 4500, 4900, 5292, 5400, 6084, 6300, 7056, 7200, 8100, 8712, 8820, 9000, 9800, 9900, 10404, 10584, 10800, 11025, 11700, 12100, 12168, 12348, 12600, 12996, 13068, 13500, 14112, 14400, 14700, 15300, 15876, 16200, 16900
OFFSET
1,1
COMMENTS
From Amiram Eldar, Nov 01 2020: (Start)
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.
The complement of the union of A005117, A190641 and A338539.
The asymptotic density of this sequence is 1 - 6/Pi^2 - (6/Pi^2)*A154945 - (3/Pi^2)*(A154945^2 - A324833) = 0.0033907041... (End)
LINKS
EXAMPLE
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:
1800 = (6*15*20) = (10*12*15).
9900 = (6*10*165) = (6*15*110) = (10*15*66).
5400 = (6*20*45) = (10*12*45) = (10*15*36) = (15*18*20).
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.
MATHEMATICA
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@@# &]] &] != {}]]
Select[Range[20000], Count[FactorInteger[#][[;; , 2]], _?(#1 > 1 &)] > 2 &] (* Amiram Eldar, Nov 01 2020 *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 02 2018
STATUS
approved