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Old 2021-08-10, 05:01   #12
Happy5214
 
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"Alexander"
Nov 2008
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Quote:
Originally Posted by henryzz View Post
The following may or may not be optimal(likely not)
106 / 2
53 - 1
52 / 4
13 - 1
12 / 3
4 / 2
Quote:
Originally Posted by Viliam Furik View Post
I think this is the best you can do for this number.
Tied for the best?

106 / 2
53 + 1
54 / 2
27 / 3
9 / 3
3 - 1
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Old 2021-08-10, 07:25   #13
axn
 
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Quote:
Originally Posted by Happy5214 View Post
Tied for the best?

106 / 2
53 + 1
54 / 2
27 / 3
9 / 3
3 - 1
106 + 1
107 + 1
108 / 9
12 / 3
4 / 2

106 - 1
105 / 7 (alternate 105 / 3 / 5 + 1 / 2 / 2)
15 / 3 ; 15 + 1
5 - 1 ; 16 / 4
4 / 2
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Old 2021-08-10, 11:43   #14
R. Gerbicz
 
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"Robert Gerbicz"
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Quote:
Originally Posted by ThomasK View Post
The best algorithms for the minimum total analysis up to n have the running time O (n * ln n) and the memory requirement O(n0.5).
Time in O(n*loglog(n)) is also possible, because with easy proof you can assume that you can always choose d=p prime, where p^2<=n.

When you compute the sequence:
W(n)<=min{W(n/p): p|n and p^2<=n}.
Calculate W() in large blocks, say in [1+2^e,2^(e+1)] then we already know the n/p<=n/2<=2^e W values.

W(n)<=min(W[n-1]+1,W[n+1]+1), but we wouldn't use +1 after -1, so with two pass you can get the whole W() sequence, once go from left then right.

W(n)<log2(n) is also trivial (use binary expansion of n: repeatedly divide by 2 for even n, otherwise subtract one).

With slow factoring, but the obvious sieving method gives the O(n*loglog(n)) algorithm. Use PARI-Gp,
Code:
F(E)={W=vector(2^E,n,E);
W[1]=W[2]=0;
for(e=1,E-1,for(n=1+2^e,2^(e+1),
a=factor(n);o=matsize(a)[1];
for(i=1,o,p=a[i,1];if(p^2<=n,W[n]=min(W[n],W[n/p]))));
for(n=1+2^e,2^(e+1),W[n]=min(W[n],W[n-1]+1));
forstep(n=-1+2^(e+1),1+2^e,-1,W[n]=min(W[n],W[n+1]+1)));
return(W)}

F(20);

for(h=0,5,print(h"   "sum(i=2,10^6,W[i]==h)))
The counts for [2,10^6] if you could check: [the counts for the given [2,1000] is good]
Code:
W() count
0   48474
1   614400
2   328988
3   8137
4   0
5   0

Last fiddled with by R. Gerbicz on 2021-08-10 at 11:45
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Old 2021-08-10, 12:07   #15
Viliam Furik
 
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Quote:
Originally Posted by Happy5214 View Post
Tied for the best?

106 / 2
53 + 1
54 / 2
27 / 3
9 / 3
3 - 1
I messed up. I forgot you can do +1.
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Old 2021-08-10, 12:58   #16
ThomasK
 
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To give you a feeling how the distribution of the numbers is developing, I give you the function values for all powers of ten from 1 to 12.


W0(10): 3
W1(10): 6
W2(10): 0
W3(10): 0
W4(10): 0


W0(100): 17
W1(100): 71
W2(100): 11
W3(100): 0
W4(100): 0


W0(103): 108
W1(103): 686
W2(103): 201
W3(103): 4
W4(103): 0


W0(104): 755
W1(104): 6598
W2(104): 2592
W3(104): 54
W4(104): 0


W0(105): 5936
W1(105): 63449
W2(105): 29916
W3(105): 698
W4(105): 0


W0(106): 48474
W1(106): 614400
W2(106): 328988
W3(106): 8137
W4(106): 0


W0(107): 406270
W1(107): 5952657
W2(107): 3550745
W3(107): 90324
W4(107): 3


W0(108): 3532031
W1(108): 58088295
W2(108): 37432690
W3(108): 946964
W4(108): 19


W0(109): 31295358
W1(109): 568932663
W2(109): 390065916
W3(109): 9705879
W4(109): 183


W0(1010): 279591668
W1(1010): 5588087493
W2(1010): 4034529147
W3(1010): 97790090
W4(1010): 1601


W0(1011): 2521429242
W1(1011): 54968844332
W2(1011): 41532029309
W3(1011): 977682518
W4(1011): 14598


W0(1012): 22996137423
W1(1012): 541664112990
W2(1012): 425608837164
W3(1012): 9730782305
W4(1012): 130117


Note: Always the sum is not 10n but 10n - 1, because W (1) is not defined.


_____________________________________________________


When we calculated a total analysis up to 1010 for the first time in the early years, we thought that, contrary to our theoretical estimates, which say that the asymptotic density for the Zweiwertzahlen is 1 and the asymptotic density for the Dreiwertzahlen is 0, there was a calculation error, because after the total analysis by the computer up to 1010 the proportion of Dreiwertzahlen increases.

That is a contradiction. To solve the problem we decided to do the total analysis up to 1012.

The relief and the cheering was very great when we read the intermediate results on the screen in the middle of the night during the calculation and the proportion of Dreiwertzahlen - as had long been expected - finally fell. If this hadn't been the case, then we could have thrown away all of our theoretical estimates.

The reason this takes so long is that the function ln ln n increases extremely slowly and it takes a very long time before the function ln ln n can assert itself against multiplicative constants.

We expect that around 1024 the Zweiwertzahlen will overtake the Einswertzahlen, i.e. W2(n) > W1(n) for all n > around 1024.

However, the Einswertzahlen offer very strong resistance against the Zweiwertzahlen. According to our extrapolations, it takes an extremely long time before the Zweiwertzahlen goes to the asyptotic density 1 and the Einswertzahlen goes to the asyptotic density 0 approach.


_____________________________________


In the next posts I will tell you something about our algorithm and prove a very important theorem from the Minimum theory.
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Old 2021-08-10, 22:02   #17
ThomasK
 
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Quote:
Originally Posted by R. Gerbicz View Post
Time in O(n*loglog(n)) is also possible, because with easy proof you can assume that you can always choose d=p prime, where p^2<=n.



(...)



Many Thanks. You are right, your algorithm is better with a running time of O (n * ln ln n) than our algorithm with a running time of O (n * ln n).

Our algorithm differentiates between a Grundblock (= basic block) and a Schiebeblock (= push block). If we want to create a total analysis up to n, then the basic block has a size of at least n0.5. The values ​​of all numbers from 2 to at least n0.5 are stored in the basic block. No further analysis is necessary in this area, as all numbers are known with their value and prime factorization. It is practical if the basic block extends from 2 to 10n for all n > = 3, since W (10n) = 0 for all n > = 3. The length of the push block is n0.5. In the total analysis up to 1012, the basic block was given from 2 to 106. The first push block ranged from 106 + 1 to 2 * 106, the second push block from 2 * 106 + 1 to 3 * 106, the nth push block from n * 10 ^ 6 + 1 to (n + 1) * 10 ^ 6 and the last push block from (106 - 1) * 106 to 106 * 106. This classification guarantees that the first number of a push block is always a Einswertzahl and the last number of a push block is always a Nullwertzahl.

Now all Nullwertzahlen in a pushblock are determined within a push block; the numbers with a value > 0 fall through a sieve.

W (n) = 0 if and only if n = p1 * p2 * p3 * p4 * p5 * ... * pk, where pi are primes with p1 <= p2 <= p3 <= p4 ...; then p1 = 2 and p1 * p2 * ... * pi> = p (i + 1) for all i < k.

In the next run, all numbers are determined within the push block that can reach a zero value number with one move. These are then the Einswertzahlen. Numbers with a value > 1 fall through a sieve.
In the next run, all numbers are determined within the push block that can reach a Einswertzahl with one move, etc.
Once all the numbers within a push block have been analyzed, the corresponding analyzes are carried out and then the next push block is analyzed.

With this algorithm you can improve the multiplicative constant if you have saved the prime numbers. However, we do not manage to be better than n * ln n, which is why your algorithm is better. But I believe that beyond O (n * ln ln n) further improvements are impossible. Depending on how many prime numbers or Nullwertzahlen are stored, the only thing that matters in the runtime is the reduction of the multiplicative constant.

__________________________

In the next post we will prove the Theorem of value restriction of sufficiently large potencies:


W (nk) = 0 for all k> = ln n / ln 2 - 1 if n is even
W (nk) = 1 for all k> = (n-1) / 2 if n is odd

Last fiddled with by ThomasK on 2021-08-10 at 22:05
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Old 2021-08-12, 13:02   #18
ThomasK
 
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Theorem of value restriction of sufficiently large potencies:

n is a natural number with n >= 2


Then applies

W (nk) = 0 for all k > = ln n / ln 2 - 1 if n is even
W (nk) = 1 for all k > = (n-1) / 2 if n is odd


Proof:

We prove this theorem specifically for n = 2 and n = 3 and n = 4 and in general for n > = 5.

W(2k) = 0 for all k = 1, 2, 3, 4, ....(Powers of 4 included)
W(3k) = 1 for all k = 1, 2, 3, 4, ....

Let n > = 5 be a natural number and k a natural number.

First case: n is even

n = 2 * (n / 2) and nk = 2k * (n/2)k. In the worst case, n / 2 is a prime number.

But if n / 2 <= 2k, you can divide each away n / 2 in the first k steps and get the Nullwertzahl 2k. A sufficient condition for nk to be a Nullwertzahl is therefore 2k > = n / 2 or k > = ln (n / 2) / ln 2 = ln n / ln 2 - 1.


Second case: n is odd

n2 - 1 = (n + 1)(n - 1) = 22 * (n + 1) / 2 * (n - 1) / 2
n4 - 1 = (n2 + 1)(n2 - 1) = (n2 + 1)(n + 1)(n - 1)

In general

n(2^j) - 1 = (n(2^(j-1)) + 1)(n(2^(j-1)) - 1) =
2(j+1) * (n(2^(j-1)) + 1) / 2 * (n(2^(j-2)) + 1) / 2 * ... * (n(2^1) + 1) / 2 * (n + 1) / 2 * (n - 1) / 2

Let us first take the case that k = 2j is a power of 2. Then we subtract one from nk and get nk - 1, which costs us a move. From the above factorization of n(2^j) - 1 we can now divide away all the odd factors one after the other, first the largest, then the second largest, and so on, because of n > = 5 it the largest factor in each case is still smaller than the product of remaining factors. The last thing left is (n - 1) / 2, which in the worst case is a prime number. So n(2^j) - 1 is a Nullwertzahl if 2(j+1) >= (n - 1) / 2, from which 2j > = (n - 1) / 4 follows.

In the general case that k is not a power of 2, we divide n from nk away until the remaining exponent is a power of 2 that is 2j. The original exponent k must shrink by half at most. So if k > = (n - 1) / 2, then 2j > = (n - 1) / 4. So it follows that W (nk) = 1 for all k > = (n - 1) / 2.


With this the Theorem of value restriction of sufficiently large potencies is completely proved!


__________________________________________________________


This theorem has far-reaching consequences in Minimum theory.

We denote the set of all Nullwertzahlen with W0, the set of all Einswertzahlen with W1, the set of all Zweiwertzahlen with W2, the set of all Dreiwertzahlen with W3, the set of all Vierwertzahlen with W4, and the set of all k-valued numbers with Wk.

If every sufficiently large power of EVERY even natural number is a Nullwertzahl and every sufficiently large power of EVERY odd natural number is a Einswertzahl, then the conjecture that the density of the set of all Numbers Wk is 0 is false!

Sometimes we call the set of Nullwertzahlen und Einswertzahlen together as cheap numbers and those numbers with W (n) > = 3 as expensive numbers. So far we have only carried out the total analysis up to 1012, but we took a random look at the distribution of the numbers above 1012. The following table shows the number of Nullwertzahlen up to the Vierwertzahlen in the first milliard above 1012, 1013 and 1014.

The distribution is as our theoretical estimates predict. But there is still a long way to go to the smallest Fünfwertzahl, which we suspect to be on the order of 1018.


W0(1012 + 109) - W0(1012): 22129699
W1(1012 + 109) - W1(1012): 538366808
W2(1012 + 109) - W2(1012): 429800315
W3(1012 + 109) - W3(1012): 9703057
W4(1012 + 109) - W4(1012): 121

W0(1013 + 109) - W0(1013): 20455311
W1(1013 + 109) - W1(1013): 531664281
W2(1013 + 109) - W2(1013): 438266115
W3(1013 + 109) - W3(1013): 9614183
W4(1013 + 109) - W4(1013): 110

W0(1014 + 109) - W0(1014): 19015775
W1(1014 + 109) - W1(1014): 525804423
W2(1014 + 109) - W2(1014): 445685370
W3(1014 + 109) - W3(1014): 9494338
W4(1014 + 109) - W4(1014): 94


In the next Posting we will compare the Nullwertzahlen with the prime numbers. The comparison of the functions W0(n) and pi (n) is very interesting.
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Old 2021-08-17, 00:34   #19
ThomasK
 
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In this post we will compare the Nullwertzahlen with the prime numbers. In particular, it is also about the comparison of the functions Pi (x) and W0(x).

Pi (x) is the number of prime numbers less than or equal to x and W0(x) is the number of Nullwertzahlen less than or equal to x.

To get a feel for why the Nullwertzahlen behave like prime numbers on the one hand, but also like inverse prime numbers on the other, here all prime numbers and all Nullwertzahlen up to 10000.


Prime Numbers up to 10000:

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, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973.



Nullwertzahlen up to 10000:

2, 4, 8, 12, 16, 24, 32, 36, 40, 48, 56, 60, 64, 72, 80, 84, 96, 108, 112, 120, 128, 132, 144, 160, 168, 176, 180, 192, 200, 208, 216, 224, 240, 252, 256, 264, 280, 288, 300, 312, 320, 324, 336, 352, 360, 384, 392, 396, 400, 408, 416, 420, 432, 440, 448, 456, 468, 480, 504, 512, 520, 528, 540, 544, 552, 560, 576, 588, 600, 608, 612, 616, 624, 640, 648, 660, 672, 680, 684, 704, 720, 728, 736, 756, 760, 768, 780, 784, 792, 800, 816, 828, 832, 840, 864, 880, 896, 900, 912, 920, 924, 928, 936, 952, 960, 972, 992, 1000, 1008, 1020, 1024, 1040, 1044, 1056, 1064, 1080, 1088, 1092, 1104, 1116, 1120, 1140, 1152, 1160, 1176, 1188, 1200, 1216, 1224, 1232, 1240, 1248, 1260, 1280, 1288, 1296, 1320, 1344, 1360, 1368, 1380, 1392, 1400, 1404, 1408, 1428, 1440, 1452, 1456, 1472, 1480, 1488, 1500, 1512, 1520, 1536, 1560, 1568, 1584, 1596, 1600, 1620, 1624, 1632, 1656, 1664, 1680, 1716, 1728, 1736, 1740, 1760, 1764, 1776, 1792, 1800, 1824, 1836, 1840, 1848, 1856, 1860, 1872, 1904, 1920, 1932, 1936, 1944, 1960, 1968, 1980, 1984, 2000, 2016, 2040, 2048, 2052, 2064, 2072, 2080, 2088, 2100, 2112, 2128, 2160, 2176, 2184, 2200, 2208, 2220, 2232, 2240, 2244, 2256, 2268, 2280, 2288, 2296, 2304, 2320, 2340, 2352, 2368, 2376, 2400, 2408, 2432, 2436, 2448, 2460, 2464, 2480, 2484, 2496, 2508, 2520, 2560, 2576, 2580, 2592, 2600, 2604, 2624, 2632, 2640, 2664, 2688, 2700, 2704, 2720, 2736, 2744, 2752, 2760, 2772, 2784, 2800, 2808, 2816, 2820, 2856, 2880, 2904, 2912, 2916, 2940, 2944, 2952, 2960, 2968, 2976, 2992, 3000, 3008, 3024, 3036, 3040, 3060, 3072, 3080, 3096, 3108, 3120, 3132, 3136, 3168, 3180, 3192, 3200, 3240, 3248, 3264, 3276, 3280, 3300, 3312, 3328, 3344, 3348, 3360, 3384, 3392, 3400, 3420, 3432, 3440, 3444, 3456, 3472, 3480, 3520, 3528, 3536, 3540, 3552, 3564, 3584, 3600, 3612, 3640, 3648, 3672, 3680, 3696, 3712, 3720, 3744, 3760, 3776, 3780, 3800, 3808, 3816, 3828, 3840, 3864, 3872, 3888, 3900, 3904, 3920, 3936, 3948, 3952, 3960, 3968, 3996, 4000, 4032, 4048, 4056, 4080, 4092, 4096, 4104, 4116, 4128, 4140, 4144, 4160, 4176, 4200, 4212, 4224, 4240, 4248, 4256, 4284, 4312, 4320, 4352, 4356, 4368, 4392, 4400, 4416, 4428, 4440, 4452, 4464, 4480, 4488, 4500, 4512, 4536, 4560, 4576, 4592, 4600, 4608, 4620, 4640, 4644, 4680, 4704, 4720, 4736, 4752, 4760, 4784, 4788, 4800, 4816, 4824, 4840, 4860, 4864, 4872, 4880, 4884, 4896, 4920, 4928, 4956, 4960, 4968, 4992, 5000, 5016, 5040, 5076, 5088, 5096, 5100, 5104, 5112, 5120, 5124, 5148, 5152, 5160, 5184, 5200, 5208, 5220, 5248, 5264, 5280, 5292, 5304, 5320, 5328, 5360, 5376, 5400, 5408, 5412, 5440, 5456, 5460, 5472, 5488, 5504, 5508, 5520, 5544, 5568, 5580, 5600, 5616, 5628, 5632, 5640, 5664, 5676, 5680, 5700, 5712, 5720, 5724, 5760, 5796, 5800, 5808, 5824, 5832, 5840, 5856, 5880, 5888, 5904, 5920, 5928, 5936, 5940, 5952, 5964, 5984, 6000, 6016, 6032, 6048, 6072, 6080, 6084, 6120, 6132, 6144, 6156, 6160, 6192, 6200, 6204, 6216, 6240, 6264, 6272, 6300, 6320, 6336, 6360, 6372, 6384, 6400, 6432, 6440, 6448, 6468, 6480, 6496, 6512, 6528, 6552, 6560, 6588, 6600, 6608, 6624, 6636, 6656, 6660, 6664, 6688, 6696, 6720, 6732, 6760, 6768, 6776, 6784, 6800, 6804, 6816, 6832, 6840, 6864, 6880, 6888, 6900, 6912, 6936, 6944, 6960, 6972, 6996, 7000, 7008, 7020, 7040, 7056, 7072, 7080, 7104, 7128, 7140, 7168, 7176, 7200, 7216, 7224, 7236, 7260, 7280, 7296, 7308, 7320, 7344, 7360, 7380, 7392, 7400, 7424, 7440, 7448, 7452, 7480, 7488, 7500, 7504, 7520, 7524, 7552, 7560, 7568, 7584, 7600, 7616, 7632, 7644, 7656, 7668, 7680, 7696, 7728, 7740, 7744, 7752, 7776, 7788, 7800, 7808, 7812, 7840, 7872, 7884, 7896, 7904, 7920, 7936, 7952, 7956, 7968, 7980, 7992, 8000, 8008, 8040, 8052, 8064, 8096, 8100, 8112, 8120, 8160, 8176, 8184, 8192, 8200, 8208, 8232, 8256, 8272, 8280, 8288, 8316, 8320, 8352, 8360, 8400, 8424, 8448, 8460, 8480, 8496, 8512, 8520, 8528, 8532, 8544, 8568, 8576, 8580, 8600, 8624, 8640, 8664, 8680, 8700, 8704, 8712, 8736, 8748, 8760, 8784, 8800, 8820, 8832, 8840, 8844, 8848, 8856, 8880, 8892, 8904, 8928, 8944, 8960, 8964, 8976, 9000, 9016, 9024, 9048, 9072, 9088, 9108, 9120, 9152, 9180, 9184, 9200, 9216, 9240, 9248, 9280, 9288, 9296, 9300, 9324, 9328, 9344, 9360, 9372, 9384, 9396, 9400, 9408, 9440, 9464, 9472, 9480, 9504, 9520, 9540, 9568, 9576, 9600, 9612, 9632, 9636, 9648, 9660, 9672, 9680, 9720, 9728, 9744, 9760, 9768, 9776, 9792, 9800, 9828, 9840, 9856, 9880, 9900, 9912, 9920, 9936, 9960, 9968, 9984, 9996, 10000.



If you look at the distribution of the prime numbers as well as the distribution of the Nullwertzahlen, you will see amazing similarities on the one hand, but also some interesting differences on the other.


Number of divisors:

Prime number: 2
Nullwertzahl: at least C * ln n
___________________________________

Number of prim divisors:

Prime number: 1
Nullwertzahl: at least C * ln ln n
___________________________________

Construction:

Prim number: very difficult
Nullwertzahl: very easy
____________________________________

Determination of the prime divisors:

Prim number: very easy
Nullwertzahl: difficult
______________________________________

Congruence without the number 2:

Prim number: 1 mod 4 or 3 mod 4
Nullwertzahl: 0 mod 4
________________________________________

Difference of twins:

Prim number: 2
Nullwertzahl : 4
________________________________________

Construction of twins:

Prim number (p is prime und p + 2 is also prime): very difficult
Nullwertzahl (n is Nullwertzahl and n + 4 is also Nullwertzahl): very difficult
_________________________________________

Number of twins up to n:

Prim number: presumably O(n / (ln n)2)
Nullwertzahlen: presumably O(n / (ln n)2)
_____________________________________________

Distribution pattern locally:

Prim number: chaotic statistical
Nullwertzahlen: chaotic statistical plus Chaos surcharge for self-similarity reflection
____________________________________________

Distribution pattern global:

Prim number: regularly
Nullwertzahlen: regularly with greater fuzziness because of Chaos surcharge for self-similarity reflection
____________________________________________

Maximum distance between two immediate neighbors:

Prim number: presumably O((ln n)2)
Nullwertzahl: presumably O((ln n)2)


Note:

Even if the Riemann hypothesis, which has not yet been proven, should apply, it can be assumed that it can only be proven that the maximum distance between two immediately adjacent prime numbers in the number range up to n is at most O (n0.5), since the zeros of the continuation of the Riemannian Zeta function with real part 0.5 have a statistical background noise of C * n ^ 0.5 / ln n and also infinitely often for every constant C, however large, both R (x) - pi (x)> C * x [SUP] 0.5 [/ SUP] / ln x and pi (x) - R (x)> C * x [SUP] 0.5 [/ SUP] / ln x is forced. For the Nullwertzahlen plus the chaos surcharge for self-similarity reflection!


Direct comparison of the functions W0(n) and Pi (n) for the powers of ten from 1 to 12:


W0(10) = 3
Pi(10) = 4
W0(10) / Pi(10) = 0.75
_____________________________________________

W0(100) = 17
Pi(100) = 25
W0(100) / Pi(100) = 0.68
_____________________________________________

W0(103) = 108
Pi(103) = 168
W0(103) / Pi(103) = 0.642857...
_____________________________________________

W0(104) = 755
Pi(104) = 1229
W0(104) / Pi(104) = 0.61432...
_____________________________________________

W0(105) = 5936
Pi(105) = 9592
W0(105) / Pi(105) = 0.618849...
_____________________________________________

W0(106) = 48474
Pi(106) = 78498
W0(106) / Pi(106) = 0.6175...
_____________________________________________

W0(107) = 406270
Pi(107) = 664579
W0(107) / Pi(107) = 0.611319...
_____________________________________________

W0(108) = 3532031
Pi(108) = 5761455
W0(108) / Pi(108) = 0.61304...
_____________________________________________

W0(109) = 31295358
Pi(109) = 50847534
W0(109) / Pi(109) = 0.61547...
_____________________________________________

W0(1010) = 279591668
Pi(1010) = 455052511
W0(1010) / Pi(1010) = 0.6144...
_____________________________________________

W0(1011) = 2521429242
Pi(1011) = 4118054813
W0(1011) / Pi(108) = 0.612286...
_____________________________________________

W0(1012) = 22996137423
Pi(1012) = 37607912018
W0(1012) / Pi(1012) = 0.61147....
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Old 2021-08-18, 08:55   #20
ThomasK
 
Aug 2021

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You can find our last article on Minimum at the following link, Page 57 - 66:


https://moodle.phst.at/pluginfile.ph...201502_ges.pdf


The article is in German, but many things should be understandable even without knowledge of the German language.
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Old 2021-10-19, 14:47   #21
arbooker
 
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"Andrew Booker"
Mar 2013

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Interesting game. Maybe you know this already, but there is a simple proof that the Nullwertzahlen have density 0: any Nullwertzahl \(n\) satisfies \(n\le2^{2^{\Omega(n)-1}}\), so that \(\Omega(n)>\log\log(n)/\log(2)\). As you pointed out in another post, \(\Omega(n)\) is almost always close to \(\log\log{n}\), so very few \(n\) have so many prime factors. Following the proof of the Hardy-Ramanujan theorem, I think you can push this as far as a proof that \(W_0(x)=O(x/(\log{x})^\delta)\) for some \(\delta>0\), but I don't see how to get \(\delta=1\) from just this.
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