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A190879 Numbers k having at least three distinct prime divisors and being divisible by the square of the sum of their prime divisors. 1
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..54.

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] (* Jean-Fran├žois Alcover, Sep 23 2011 *)

CROSSREFS

Sequence in context: A116346 A102509 A250008 * A154061 A253650 A054026

Adjacent sequences:  A190876 A190877 A190878 * A190880 A190881 A190882

KEYWORD

nonn

AUTHOR

Michel Lagneau, May 23 2011

EXTENSIONS

Definition modified by Harvey P. Dale, Oct 12 2014

STATUS

approved

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Last modified October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)