A190879 Numbers k having at least three distinct prime divisors and being divisible by the square of the sum of their prime divisors.

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

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

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

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-François Alcover, Sep 23 2011 *)

Crossrefs

Sequence in context: A102509 A250008 A332282 * 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