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 Take Divisors, Get Square Root (Posted on 2007-07-13)
Determine all positive whole numbers w such that d(w) = √w, where d(w) denotes the number of positive divisors of w.

 See The Solution Submitted by K Sengupta Rating: 4.0000 (2 votes)

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 computer exploration--spoiler | Comment 1 of 5

5   dim FctCt(33)
10   for N=1 to 1000000
20    X=N
25    FacNo=0:PrevFctr=0
30    repeat
40      Pfctr=prmdiv(X):X=X//Pfctr
50      if Pfctr=PrevFctr then
60       :FctCt(FacNo)=FctCt(FacNo)+1
70      :else
80       :FacNo=FacNo+1
90       :FctCt(FacNo)=1
95      PrevFctr=Pfctr
100    until Pfctr=1
110    NDivsr=1
120    for I=1 to FacNo
130     NDivsr=NDivsr*(FctCt(I)+1)
140    next
145    NDivsr=NDivsr//2
150    if NDivsr*NDivsr=N then print N,NDivsr
200   next

finds only 1 and 9, where 1 has only 1 divisor, itself, and 9 has 3 divisors: 1, 3 and 9.

The program did check all numbers up to 1,000,000.

What's needed is a proof that above some value there, the number of divisors will always be below the square root of the number.

 Posted by Charlie on 2007-07-13 11:14:19
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