

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
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OFFSET

1,1


COMMENTS

From Amiram Eldar, Nov 01 2020: (Start)
Also, numbers with more than two nonunitary 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

Amiram Eldar, Table of n, a(n) for n = 1..10000


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 *)


CROSSREFS

Cf. A001055, A001221, A001222, A007716, A045778, A051185, A078374, A281116, A303140, A303283, A305843, A305854, A317748, A318715, A318717, A318721.
Cf. A005117, A036785, A056170, A154945, A190641, A324833, A338539.
Sequence in context: A344595 A344694 A127658 * A338540 A137490 A129575
Adjacent sequences: A318717 A318718 A318719 * A318721 A318722 A318723


KEYWORD

nonn


AUTHOR

Gus Wiseman, Sep 02 2018


STATUS

approved



