20210810, 05:01  #12 
"Alexander"
Nov 2008
The Alamo City
13·71 Posts 

20210810, 07:25  #13 
Jun 2003
2·2,719 Posts 

20210810, 11:43  #14  
"Robert Gerbicz"
Oct 2005
Hungary
1615_{10} Posts 
Quote:
When you compute the sequence: W(n)<=min{W(n/p): pn 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[n1]+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 PARIGp, Code:
F(E)={W=vector(2^E,n,E); W[1]=W[2]=0; for(e=1,E1,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[n1]+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))) Code:
W() count 0 48474 1 614400 2 328988 3 8137 4 0 5 0 Last fiddled with by R. Gerbicz on 20210810 at 11:45 

20210810, 12:07  #15 
"Viliam Furík"
Jul 2018
Martin, Slovakia
13×61 Posts 

20210810, 12:58  #16 
Aug 2021
13_{8} Posts 
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.
W_{0}(10): 3 W_{1}(10): 6 W_{2}(10): 0 W_{3}(10): 0 W_{4}(10): 0 W_{0}(100): 17 W_{1}(100): 71 W_{2}(100): 11 W_{3}(100): 0 W_{4}(100): 0 W_{0}(10^{3}): 108 W_{1}(10^{3}): 686 W_{2}(10^{3}): 201 W_{3}(10^{3}): 4 W_{4}(10^{3}): 0 W_{0}(10^{4}): 755 W_{1}(10^{4}): 6598 W_{2}(10^{4}): 2592 W_{3}(10^{4}): 54 W_{4}(10^{4}): 0 W_{0}(10^{5}): 5936 W_{1}(10^{5}): 63449 W_{2}(10^{5}): 29916 W_{3}(10^{5}): 698 W_{4}(10^{5}): 0 W_{0}(10^{6}): 48474 W_{1}(10^{6}): 614400 W_{2}(10^{6}): 328988 W_{3}(10^{6}): 8137 W_{4}(10^{6}): 0 W_{0}(10^{7}): 406270 W_{1}(10^{7}): 5952657 W_{2}(10^{7}): 3550745 W_{3}(10^{7}): 90324 W_{4}(10^{7}): 3 W_{0}(10^{8}): 3532031 W_{1}(10^{8}): 58088295 W_{2}(10^{8}): 37432690 W_{3}(10^{8}): 946964 W_{4}(10^{8}): 19 W_{0}(10^{9}): 31295358 W_{1}(10^{9}): 568932663 W_{2}(10^{9}): 390065916 W_{3}(10^{9}): 9705879 W_{4}(10^{9}): 183 W_{0}(10^{10}): 279591668 W_{1}(10^{10}): 5588087493 W_{2}(10^{10}): 4034529147 W_{3}(10^{10}): 97790090 W_{4}(10^{10}): 1601 W_{0}(10^{11}): 2521429242 W_{1}(10^{11}): 54968844332 W_{2}(10^{11}): 41532029309 W_{3}(10^{11}): 977682518 W_{4}(10^{11}): 14598 W_{0}(10^{12}): 22996137423 W_{1}(10^{12}): 541664112990 W_{2}(10^{12}): 425608837164 W_{3}(10^{12}): 9730782305 W_{4}(10^{12}): 130117 Note: Always the sum is not 10^{n} but 10^{n}  1, because W (1) is not defined. _____________________________________________________ When we calculated a total analysis up to 10^{10} 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 10^{10} the proportion of Dreiwertzahlen increases. That is a contradiction. To solve the problem we decided to do the total analysis up to 10^{12}. 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 10^{24} the Zweiwertzahlen will overtake the Einswertzahlen, i.e. W_{2}(n) > W_{1}(n) for all n > around 10^{24}. 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. 
20210810, 22:02  #17  
Aug 2021
1011_{2} Posts 
Quote:
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 n^{0.5}. The values of all numbers from 2 to at least n^{0.5} 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 10^{n} for all n > = 3, since W (10^{n}) = 0 for all n > = 3. The length of the push block is n^{0.5}. In the total analysis up to 10^{12}, the basic block was given from 2 to 10^{6}. The first push block ranged from 10^{6} + 1 to 2 * 10^{6}, the second push block from 2 * 10^{6} + 1 to 3 * 10^{6}, the nth push block from n * 10 ^ 6 + 1 to (n + 1) * 10 ^ 6 and the last push block from (10^{6}  1) * 10^{6} to 10^{6} * 10^{6}. 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 (n^{k}) = 0 for all k> = ln n / ln 2  1 if n is even W (n^{k}) = 1 for all k> = (n1) / 2 if n is odd Last fiddled with by ThomasK on 20210810 at 22:05 

20210812, 13:02  #18 
Aug 2021
11 Posts 
Theorem of value restriction of sufficiently large potencies:
n is a natural number with n >= 2 Then applies W (n^{k}) = 0 for all k > = ln n / ln 2  1 if n is even W (n^{k}) = 1 for all k > = (n1) / 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(2^{k}) = 0 for all k = 1, 2, 3, 4, ....(Powers of 4 included) W(3^{k}) = 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 n^{k} = 2^{k} * (n/2)^{k}. In the worst case, n / 2 is a prime number. But if n / 2 <= 2^{k}, you can divide each away n / 2 in the first k steps and get the Nullwertzahl 2^{k}. A sufficient condition for n^{k} to be a Nullwertzahl is therefore 2^{k} > = n / 2 or k > = ln (n / 2) / ln 2 = ln n / ln 2  1. Second case: n is odd n^{2}  1 = (n + 1)(n  1) = 2^{2} * (n + 1) / 2 * (n  1) / 2 n^{4}  1 = (n^{2} + 1)(n^{2}  1) = (n^{2} + 1)(n + 1)(n  1) In general n^{(2^j)}  1 = (n^{(2^(j1))} + 1)(n^{(2^(j1))}  1) = 2^{(j+1)} * (n^{(2^(j1))} + 1) / 2 * (n^{(2^(j2))} + 1) / 2 * ... * (n^{(2^1)} + 1) / 2 * (n + 1) / 2 * (n  1) / 2 Let us first take the case that k = 2^{j} is a power of 2. Then we subtract one from n^{k} and get n^{k}  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 2^{j} > = (n  1) / 4 follows. In the general case that k is not a power of 2, we divide n from n^{k} away until the remaining exponent is a power of 2 that is 2^{j}. The original exponent k must shrink by half at most. So if k > = (n  1) / 2, then 2^{j} > = (n  1) / 4. So it follows that W (n^{k}) = 1 for all k > = (n  1) / 2. With this the Theorem of value restriction of sufficiently large potencies is completely proved! __________________________________________________________ This theorem has farreaching consequences in Minimum theory. We denote the set of all Nullwertzahlen with W_{0}, the set of all Einswertzahlen with W_{1}, the set of all Zweiwertzahlen with W_{2}, the set of all Dreiwertzahlen with W_{3}, the set of all Vierwertzahlen with W_{4}, and the set of all kvalued numbers with W_{k}. 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 W_{k} 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 10^{12}, but we took a random look at the distribution of the numbers above 10^{12}. The following table shows the number of Nullwertzahlen up to the Vierwertzahlen in the first milliard above 10^{12}, 10^{13} and 10^{14}. 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 10^{18}. W_{0}(10^{12} + 10^{9})  W_{0}(10^{12}): 22129699 W_{1}(10^{12} + 10^{9})  W_{1}(10^{12}): 538366808 W_{2}(10^{12} + 10^{9})  W_{2}(10^{12}): 429800315 W_{3}(10^{12} + 10^{9})  W_{3}(10^{12}): 9703057 W_{4}(10^{12} + 10^{9})  W_{4}(10^{12}): 121 W_{0}(10^{13} + 10^{9})  W_{0}(10^{13}): 20455311 W_{1}(10^{13} + 10^{9})  W_{1}(10^{13}): 531664281 W_{2}(10^{13} + 10^{9})  W_{2}(10^{13}): 438266115 W_{3}(10^{13} + 10^{9})  W_{3}(10^{13}): 9614183 W_{4}(10^{13} + 10^{9})  W_{4}(10^{13}): 110 W_{0}(10^{14} + 10^{9})  W_{0}(10^{14}): 19015775 W_{1}(10^{14} + 10^{9})  W_{1}(10^{14}): 525804423 W_{2}(10^{14} + 10^{9})  W_{2}(10^{14}): 445685370 W_{3}(10^{14} + 10^{9})  W_{3}(10^{14}): 9494338 W_{4}(10^{14} + 10^{9})  W_{4}(10^{14}): 94 In the next Posting we will compare the Nullwertzahlen with the prime numbers. The comparison of the functions W_{0}(n) and pi (n) is very interesting. 
20210817, 00:34  #19 
Aug 2021
11 Posts 
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 W_{0}(x).
Pi (x) is the number of prime numbers less than or equal to x and W_{0}(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 selfsimilarity reflection ____________________________________________ Distribution pattern global: Prim number: regularly Nullwertzahlen: regularly with greater fuzziness because of Chaos surcharge for selfsimilarity 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 (n^{0.5}), 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 selfsimilarity reflection! Direct comparison of the functions W_{0}(n) and Pi (n) for the powers of ten from 1 to 12: W_{0}(10) = 3 Pi(10) = 4 W_{0}(10) / Pi(10) = 0.75 _____________________________________________ W_{0}(100) = 17 Pi(100) = 25 W_{0}(100) / Pi(100) = 0.68 _____________________________________________ W_{0}(10^{3}) = 108 Pi(10^{3}) = 168 W_{0}(10^{3}) / Pi(10^{3}) = 0.642857... _____________________________________________ W_{0}(10^{4}) = 755 Pi(10^{4}) = 1229 W_{0}(10^{4}) / Pi(10^{4}) = 0.61432... _____________________________________________ W_{0}(10^{5}) = 5936 Pi(10^{5}) = 9592 W_{0}(10^{5}) / Pi(10^{5}) = 0.618849... _____________________________________________ W_{0}(10^{6}) = 48474 Pi(10^{6}) = 78498 W_{0}(10^{6}) / Pi(10^{6}) = 0.6175... _____________________________________________ W_{0}(10^{7}) = 406270 Pi(10^{7}) = 664579 W_{0}(10^{7}) / Pi(10^{7}) = 0.611319... _____________________________________________ W_{0}(10^{8}) = 3532031 Pi(10^{8}) = 5761455 W_{0}(10^{8}) / Pi(10^{8}) = 0.61304... _____________________________________________ W_{0}(10^{9}) = 31295358 Pi(10^{9}) = 50847534 W_{0}(10^{9}) / Pi(10^{9}) = 0.61547... _____________________________________________ W_{0}(10^{10}) = 279591668 Pi(10^{10}) = 455052511 W_{0}(10^{10}) / Pi(10^{10}) = 0.6144... _____________________________________________ W_{0}(10^{11}) = 2521429242 Pi(10^{11}) = 4118054813 W_{0}(10^{11}) / Pi(10^{8}) = 0.612286... _____________________________________________ W_{0}(10^{12}) = 22996137423 Pi(10^{12}) = 37607912018 W_{0}(10^{12}) / Pi(10^{12}) = 0.61147.... 
20210818, 08:55  #20 
Aug 2021
11 Posts 
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. 
20211019, 14:47  #21 
"Andrew Booker"
Mar 2013
2^{2}×23 Posts 
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 HardyRamanujan 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.

20211101, 23:11  #22  
Aug 2021
11 Posts 
Quote:
Thanks very much. That's right; a randomly drawn natural number n has approximately ln ln n + O (1) prime factors, where the standard deviation is O ((ln ln n) ^ 0.5). The number of all natural numbers less than or equal to n that have exactly k prime factors is asyptotically n * (ln ln n) ^ (k1) / ((ln n) * (k1)!) For k = 1, the prime number theorem results as a special case, which says that the number of prime numbers less than or equal to n is asymptotically about n / ln n. For k = 2, all natural numbers are counted with exactly 2 prime factors, e.g. 4453 = 61 * 73 is a semiprime number. The number of all semiprime numbers less than or equal to n is thus asymptotically n * ln ln n / ln n. The number of all natural numbers less than or equal to n that have exactly 3 prime factors is thus asymptotically n * (ln ln n) ^ 2 / (ln n * 2) The number of all natural numbers less than or equal to n that have exactly 4 prime factors is thus asymptotically n * (ln ln n) ^ 3 / (ln n * 6) The number of all natural numbers less than or equal to n that have exactly 5 prime factors is thus asymptotically n * (ln ln n) ^ 4 / (ln n * 24) etc. _________________________ We are now considering extending the minimum total analysis, which was previously performed to 10 ^ 12, to 10 ^ 17. We do not yet believe that we will reach the smallest Fünfwertzahl (fivevalued number) that we expect at around 10 ^ 18. Nevertheless, the minimum total analysis up to 10^17 would enable us to make a significantly improved extrapolation. Anyone who has ideas on how to efficiently carry out a minimum total analysis that runs on 1000 processors at the same time, for example, can post his ideas in this thread. How big should the basic block be? How can the algorithm be efficiently parallelized? Which programming language is best suited for the minimum total analysis up to 10^17? Should the minimum total analysis be calculated up to 10^17 in a cloud? How many and, if applicable, which prime numbers should be stored for the minimum total analysis up to 10^17? 

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