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%I #17 Oct 23 2019 10:06:59
%S 300,600,900,980,1008,1200,1500,1575,1800,1960,2016,2400,2700,3000,
%T 3024,3600,3920,4032,4212,4500,4725,4800,4851,4900,5200,5400,6000,
%U 6048,6860,7056,7200,7436,7500,7840,7875,8064,8100,8424,8448,9000,9072,9600,9800,10400,10800,10944,11025,12000,12096,12636,13500,13720,14112,14175
%N Numbers k having at least three distinct prime divisors and being divisible by the square of the sum of their prime divisors.
%C The reference considers the sequence {37026, 74052, 81900, ....} with the numbers having at least 4 distinct prime divisors. If k contains two prime divisors only, then k = (p^a)*(q^b), where p and q are two prime distinct divisors, and (p+q)^2 | k => p+q ==0 (mod p) or 0 (mod q), but p==0 (mod q) or q==0 (mod p) is impossible.
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 37026, p. 224, Ellipses,
%D Paris 2008.
%H Amiram Eldar, <a href="/A190879/b190879.txt">Table of n, a(n) for n = 1..10000</a>
%e 1575 is in the sequence because the distinct prime divisors of 1575 are {3, 5, 7} and
%e (3 + 5 + 7)^2 = 225, and 1575 = 225*7.
%p with(numtheory):for n from 1 to 20000 do:x:=factorset(n):n1:=nops(x):s:=0:for
%p p from 1 to n1 do: s:=s+x[p]:od:s:=s^2:if n1 >= 2 and irem(n,s)=0 then printf(`%d,
%p `,n):else fi:od:
%t ok[k_] := With[{pp = FactorInteger[k][[All, 1]]}, Length[pp] >= 3 && Divisible[k, Total[pp]^2]]; Select[ Range[15000], ok] (* _Jean-François Alcover_, Sep 23 2011 *)
%K nonn
%O 1,1
%A _Michel Lagneau_, May 23 2011
%E Definition modified by _Harvey P. Dale_, Oct 12 2014
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