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Old Apr 8th, 2006, 10:12 AM   #1
Sane
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Primes...
Is x, an element of the natural number set, more often divisible by...

p <= x/2 where p is an element of the prime number set

Or...

i <= x/2 where i is an element of the natural number set incrementing by 2 (starting at 1)?
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Old Apr 8th, 2006, 11:59 AM   #2
Dietrich
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What does Python say?
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I looked it up on the Intergnats!
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Old Apr 8th, 2006, 12:09 PM   #3
Sane
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Errm. Well I thought someone might know, because it could be an important piece of information for creating an efficient prime growth function. If I must, I'll code up a little solution then...

Edit:

(Toggle Plain Text) 
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

tres1 = 0.0
tres2 = 0.0

for i in range(100, 200):
    idiv2 = i/2

    r1 = 0
    max1 = 0
    for p1 in primes:
        max1 += 1
        if not i%p1:
            r1 += 1
        if p1 > idiv2:
            break

    r2 = 0
    max2 = 0
    p2 = 1
    while p2 <= idiv2:
        p2 += 2
        max2 += 1
        if not i%p2:
            r2 += 1

    res1 = float(r1)/float(max1)
    res2 = float(r2)/float(max2)

    tres1 += res1
    tres2 += res2
    
    print "[%s] Primes     %s\n      Naturals   %s"%(i, res1, res2)


print '\nAverage Primes     %s\n        Naturals   %s'%(tres1/99.0, tres2/99.0)


Results
(Toggle Plain Text) 
[100] Primes     0.125
      Naturals   0.08
[101] Primes     0.0
      Naturals   0.0
[102] Primes     0.1875
      Naturals   0.115384615385
[103] Primes     0.0
      Naturals   0.0
[104] Primes     0.125
      Naturals   0.0384615384615
[105] Primes     0.1875
      Naturals   0.230769230769
[106] Primes     0.117647058824
      Naturals   0.037037037037
[107] Primes     0.0
      Naturals   0.0
[108] Primes     0.117647058824
      Naturals   0.111111111111
[109] Primes     0.0
      Naturals   0.0
[110] Primes     0.176470588235
      Naturals   0.107142857143
[111] Primes     0.117647058824
      Naturals   0.0714285714286
[112] Primes     0.117647058824
      Naturals   0.0357142857143
[113] Primes     0.0
      Naturals   0.0
[114] Primes     0.176470588235
      Naturals   0.103448275862
[115] Primes     0.117647058824
      Naturals   0.0689655172414
[116] Primes     0.117647058824
      Naturals   0.0344827586207
[117] Primes     0.117647058824
      Naturals   0.137931034483
[118] Primes     0.111111111111
      Naturals   0.0333333333333
[119] Primes     0.111111111111
      Naturals   0.0666666666667
[120] Primes     0.166666666667
      Naturals   0.1
[121] Primes     0.0555555555556
      Naturals   0.0333333333333
[122] Primes     0.105263157895
      Naturals   0.0322580645161
[123] Primes     0.105263157895
      Naturals   0.0645161290323
[124] Primes     0.105263157895
      Naturals   0.0322580645161
[125] Primes     0.0526315789474
      Naturals   0.0645161290323
[126] Primes     0.157894736842
      Naturals   0.15625
[127] Primes     0.0
      Naturals   0.0
[128] Primes     0.0526315789474
      Naturals   0.0
[129] Primes     0.105263157895
      Naturals   0.0625
[130] Primes     0.157894736842
      Naturals   0.0909090909091
[131] Primes     0.0
      Naturals   0.0
[132] Primes     0.157894736842
      Naturals   0.0909090909091
[133] Primes     0.105263157895
      Naturals   0.0606060606061
[134] Primes     0.1
      Naturals   0.0294117647059
[135] Primes     0.1
      Naturals   0.176470588235
[136] Primes     0.1
      Naturals   0.0294117647059
[137] Primes     0.0
      Naturals   0.0
[138] Primes     0.15
      Naturals   0.0857142857143
[139] Primes     0.0
      Naturals   0.0
[140] Primes     0.15
      Naturals   0.0857142857143
[141] Primes     0.1
      Naturals   0.0571428571429
[142] Primes     0.0952380952381
      Naturals   0.0277777777778
[143] Primes     0.0952380952381
      Naturals   0.0555555555556
[144] Primes     0.0952380952381
      Naturals   0.0555555555556
[145] Primes     0.0952380952381
      Naturals   0.0555555555556
[146] Primes     0.0909090909091
      Naturals   0.027027027027
[147] Primes     0.0909090909091
      Naturals   0.108108108108
[148] Primes     0.0909090909091
      Naturals   0.027027027027
[149] Primes     0.0
      Naturals   0.0
[150] Primes     0.136363636364
      Naturals   0.131578947368
[151] Primes     0.0
      Naturals   0.0
[152] Primes     0.0909090909091
      Naturals   0.0263157894737
[153] Primes     0.0909090909091
      Naturals   0.105263157895
[154] Primes     0.136363636364
      Naturals   0.0769230769231
[155] Primes     0.0909090909091
      Naturals   0.0512820512821
[156] Primes     0.136363636364
      Naturals   0.0769230769231
[157] Primes     0.0
      Naturals   0.0
[158] Primes     0.0869565217391
      Naturals   0.025
[159] Primes     0.0869565217391
      Naturals   0.05
[160] Primes     0.0869565217391
      Naturals   0.025
[161] Primes     0.0869565217391
      Naturals   0.05
[162] Primes     0.0869565217391
      Naturals   0.0975609756098
[163] Primes     0.0
      Naturals   0.0
[164] Primes     0.0869565217391
      Naturals   0.0243902439024
[165] Primes     0.130434782609
      Naturals   0.146341463415
[166] Primes     0.0833333333333
      Naturals   0.0238095238095
[167] Primes     0.0
      Naturals   0.0
[168] Primes     0.125
      Naturals   0.0714285714286
[169] Primes     0.0416666666667
      Naturals   0.0238095238095
[170] Primes     0.125
      Naturals   0.0697674418605
[171] Primes     0.0833333333333
      Naturals   0.093023255814
[172] Primes     0.0833333333333
      Naturals   0.0232558139535
[173] Primes     0.0
      Naturals   0.0
[174] Primes     0.125
      Naturals   0.0681818181818
[175] Primes     0.0833333333333
      Naturals   0.0909090909091
[176] Primes     0.0833333333333
      Naturals   0.0227272727273
[177] Primes     0.0833333333333
      Naturals   0.0454545454545
[178] Primes     0.08
      Naturals   0.0222222222222
[179] Primes     0.0
      Naturals   0.0
[180] Primes     0.12
      Naturals   0.111111111111
[181] Primes     0.0
      Naturals   0.0
[182] Primes     0.12
      Naturals   0.0652173913043
[183] Primes     0.08
      Naturals   0.0434782608696
[184] Primes     0.08
      Naturals   0.0217391304348
[185] Primes     0.08
      Naturals   0.0434782608696
[186] Primes     0.12
      Naturals   0.063829787234
[187] Primes     0.08
      Naturals   0.0425531914894
[188] Primes     0.08
      Naturals   0.0212765957447
[189] Primes     0.08
      Naturals   0.127659574468
[190] Primes     0.12
      Naturals   0.0625
[191] Primes     0.0
      Naturals   0.0
[192] Primes     0.08
      Naturals   0.0208333333333
[193] Primes     0.0
      Naturals   0.0
[194] Primes     0.08
      Naturals   0.0204081632653
[195] Primes     0.12
      Naturals   0.122448979592
[196] Primes     0.08
      Naturals   0.0408163265306
[197] Primes     0.0
      Naturals   0.0
[198] Primes     0.12
      Naturals   0.1
[199] Primes     0.0
      Naturals   0.0

Average Primes     0.0856112788463
        Naturals   0.0528174032136


Well according to this, my theory was correct. But not by much. Am I missing something in my code? Or should it use a greater sample? Or maybe I should take in to consideration how far in to the iteration it finds a divisibility...
Last edited by Sane; Apr 8th, 2006 at 12:19 PM.
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