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A066423 Composite numbers n such that the product of proper divisors of the n does not equal n. 3

%I #10 Oct 18 2020 15:57:34

%S 4,9,12,16,18,20,24,25,28,30,32,36,40,42,44,45,48,49,50,52,54,56,60,

%T 63,64,66,68,70,72,75,76,78,80,81,84,88,90,92,96,98,99,100,102,104,

%U 105,108,110,112,114,116,117,120,121,124,126,128,130,132

%N Composite numbers n such that the product of proper divisors of the n does not equal n.

%C A084115(a(n))>1; complement of A084116. - _Reinhard Zumkeller_, May 12 2003

%H Harvey P. Dale, <a href="/A066423/b066423.txt">Table of n, a(n) for n = 1..2000</a>

%e The fourth composite number is 9. Its proper or aliquot divisors are 1 and 3. The product of 1 and 3 equals 3 which is not equal to 9. Therefore 9 is in the sequence.

%t Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Do[m = Composite[n]; If[ Apply[ Times, Drop[ Divisors[m], -1]] != m, Print[m]], {n, 1, 100} ]

%t Select[Range[150],CompositeQ[#]&&Times@@Most[Divisors[#]]!=#&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 18 2020 *)

%o (PARI) is(n)=my(d=numdiv(n)); d>4 || d==3 \\ _Charles R Greathouse IV_, Oct 15 2015

%Y Cf. A048741, A007422.

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Dec 26 2001

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