

A190879


Numbers k having at least three distinct prime divisors and being divisible by the square of the sum of their prime divisors.


2



300, 600, 900, 980, 1008, 1200, 1500, 1575, 1800, 1960, 2016, 2400, 2700, 3000, 3024, 3600, 3920, 4032, 4212, 4500, 4725, 4800, 4851, 4900, 5200, 5400, 6000, 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
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OFFSET

1,1


COMMENTS

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.


REFERENCES

J.M. De Koninck, Ces nombres qui nous fascinent, Entry 37026, p. 224, Ellipses,
Paris 2008.


LINKS



EXAMPLE

1575 is in the sequence because the distinct prime divisors of 1575 are {3, 5, 7} and
(3 + 5 + 7)^2 = 225, and 1575 = 225*7.


MAPLE

with(numtheory):for n from 1 to 20000 do:x:=factorset(n):n1:=nops(x):s:=0:for
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,
`, n):else fi:od:


MATHEMATICA

ok[k_] := With[{pp = FactorInteger[k][[All, 1]]}, Length[pp] >= 3 && Divisible[k, Total[pp]^2]]; Select[ Range[15000], ok] (* JeanFrançois Alcover, Sep 23 2011 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



