IDMclean Posted April 4, 2010 Report Posted April 4, 2010 Something of note, I managed to come up with the indexing method for primes only because I implicitly neglect one prime number, the 0th: 2. All primes greater than 2 are odd numbers, so they all follow the form of 2n+1. The reason for neglecting two besides it's unique status among the prime numbers comes from the way the natural number line is constructed:the number zero, 0, is the successor to no number.One, 1, is a number.Any number + 1 is a number.Analogously, The prime, 2, is the successor to no prime. [math]P_0=\{2\}[/math]The successor of any prime is of the form: [math]P_n = \{2n+1;i,j,n\in \mathbb{N}^+ \wedge n\not=2ij+i+j\}[/math] I suspect if the nonfiguratives are countably infinite, the zeroth element will have a unique form with relation to the other nonfiguratives. It'll be like zero on the natural number line, you include it as an axiom and use the 1st element with a rule to generate the rest. Btw, do we have a concise table of the first 10-20 nonfiguratives at hand for comparison? Quote
modest Posted April 4, 2010 Report Posted April 4, 2010 Non-figurate numbers:7,8,11,13,14,17,19,20,23,26,29,31,32,37,38,41,43,44,47,50,53 ,56,59,61,62,67,68,71,73,74,77,79,80,83,86,89,97,98,101,103, 104,107,109,110,113,116,119,122,127,128,131,134,137,139,140,143, 146,149,151,152,157,158,161,163,164,167,170,173,179,181,182,187, 188,191,193,194,197,199,200,203,206,209,211,212,218,221,223,224, 227,229,230,233,236,239,241,242,248,251,254,257,263,266,269,271, 272,277,278,281,283,284,290,293,296,299,302,307,308,311,313,314, 317,319,320,323,326,329,331,332,337,338,347,349,350,353,356,359, 362,367,368,371,373,374,377,379,380,383,386,389,391,392,397,398, 401,404,407,409,410,413,416,419,421,422,431,433,434,437,439,440, 443,446,449,452,457,458,461,463,464,467,470,473,476,479,482,487, 488,491,493,494,497,499,500,503,509,517,518,521,523,524,527,530, 533,536,539,541,542,547,548,551,554,557,563,566,569,571,572,577, 578,581,583,584,587,589,593,599,601,602,607,608,611,613,614,617, 619,620,623,626,629,631,632,638,641,643,644,647,649,650,653,656, 659,661,662,667,668,673,674,677,683,686,689,691,692,698,701,704, 707,709,710,713,716,719,722,727,728,731,733,734,737,739,740,743, 746,749,751,752,757,758,761,767,769,770,773,776,779,787,788,791, 794,797,799,800,803,806,809,811,812,817,818,821,823,824,827,829, 830,839,842,851,853,854,857,859,860,863,866,869,872,877,878,881, 883,884,887,890,893,896,899,901,902,907,908,911,913,914,917,919, 920,923,926,929,937,938,941,943,944,947,950,953,956,959,962,967, 968,971,974,977,979,980,983,986,989,991,992,997,998... It is OEIS--A090467. There are lists of several millions of non-figs in 20236. ~modest Quote
IDMclean Posted April 4, 2010 Report Posted April 4, 2010 Perhaps obvious, but useful to explicitly catalog the properties of the numbers when doing analysis: no nonfigurative number of the form [math]a^n[/math] (8 would invalidate this because it's of the form [math]2^3[/math]. Would restricting a to odd numbers fix the problem with this tentative rule?)No nonfigurative number ends in 5 (no odd multiples of five). Most even nonfiguratives are different from the previous and the successive even figurative by 6 and most are different from the previous nonfigurative by 1 with the exceptions occurring where there would be a number of the form a^n or 10n+5. In the below sequence, the pattern of {n+1, n+3, n+2, n+1} seems to have some kind of recursive repetition. Successive {n+3, n+3, n+3}, {n+3, n+3, n+3, n+3}, {n+3, n+3, n+3, n+3, n+3} patterns show up. Also, all nonfiguratives so far follow some kind of recursive form: {n+1, n+2, n+3, n+5, n+6, n+8}. Can we say with confidence they all follow one of those six forms or will we eventually end up with an ever growing list of forms? [math]\begin{array}{ccccccccccccccc} 7 & 8 & 11 & 13 & 14 & 17 & 19 & 20 & 23 & 26 & 29 & 31 & 32 & 37 & 38 \\ \hline n & n+1 & n+3 & n+2 & n+1 & n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+5 & n+1 & n+3 & n+2 \end{array}[/math] [math]\begin{array}{ccccccccccccccc} 41 & 43 & 44 & 47 & 50 & 53 & 56 & 59 & 61 & 62 & 67 & 68 & 71 & 73 & 74 \\ \hline n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+3 & n+3 & n+2 & n+1 & n+5 & n+1 & n+3 & n+2 & n+1 \end{array}[/math] [math]\begin{array}{ccccccccccccccc} 77 & 79 & 80 & 83 & 86 & 89 & 97 & 98 & 101 & 103 & 104 & 107 & 109 & 110 & 113 \\ \hline n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+8 & n+1 & n+3 & n+2 & n+1 & n+3 & n+2 & n+1 & n+3 \\ \end{array}[/math] [math]\begin{array}{ccccccccccccccc} 116 & 119 & 122 & 127 & 128 & 131 & 134 & 137 & 139 & 140 & 143 & 146 & 149 & 151 & 152 \\ \hline n+3 & n+3 & n+3 & n+5 & n+1 & n+3 & n+3 & n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+2 & n+1 \\ \end{array}[/math] [math]\begin{array}{ccccccccccccccc} 157 & 158 & 161 & 163 & 164 & 167 & 170 & 173 & 179 & 181 & 182 & 187 & 188 & 191 & 193 \\ \hline n+5 & n+1 & n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+6 & n+2 & n+1 & n+5 & n+1 & n+3 & n+2 \\ \end{array}[/math] [math]\begin{array}{ccccccccccccccc} 194 & 197 & 199 & 200 & 203 & 206 & 209 & 211 & 212 & 218 & 221 & 223 & 224 & 229 & 230 \\ \hline n+1 & n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+2 & n+1 & n+6 & n+3 & n+2 & n+1 & n+5 & n+1 \end{array}[/math] Does anyone else see the patterns that I do?What about the numbers 0-5? How do they fit into all this? Quote
modest Posted April 5, 2010 Report Posted April 5, 2010 IDMclean, I've answered you here, Re: Non-Figurate Numbers, where I think the thread might make the better context if that's alright (I ended up referencing a couple posts in that thread). ~modest Quote
Don Blazys Posted April 9, 2010 Author Report Posted April 9, 2010 By making adjustments to [math]k[/math], the function: [math]B(x)-B(x)*\alpha*(\mu-k)^{-1}[/math] can be made to give approximations that are almost exact for regions of [math]\varpi(x)[/math] spanning billions. Also, given recent determinations of [math]\varpi(x)[/math] we can now surmise the following: 1: Local fluctuations will eventually become inconsequential. 2: The value of [math]k[/math] rises and falls, but seems to be tending toward [math]0.[/math] However, it may be tending towards some other value such as: [math]2*e+\alpha^{\frac{1}{2}}[/math] or [math]\frac{3*\Pi}{2}[/math] in which case we may be able to determine a few extra digits of [math]\alpha[/math] by the time we get to [math]\varpi(10^{14})[/math]. 3: The function: [math]B(x)-B(x)*\alpha*(\mu-k)^{-1}[/math] will always approximate [math]\varpi(x)[/math] to a much higher degree of accuracy than [math]Li(x)[/math] approximates [math]\Pi(x)[/math]. All this is good news. :) But with all of these various "heavy duty" computing options now open to us... (See Craig D's last post.) what is our next step? :confused: Don. Quote
Don Blazys Posted April 10, 2010 Author Report Posted April 10, 2010 A "little bird" tells me that we have a: NASA Rocket Scientist / Excel Spreadsheet Wizard / Physicist / Raconteur / All Around Nice Person among us. I wonder if... Don. Quote
Don Blazys Posted April 18, 2010 Author Report Posted April 18, 2010 To Modest, Quoting Modest: Interesting stuff, Don. Thanks Modest. :) But if it weren't for Turtle, Donk and Yourself, this interesting... Heck... DOWNRIGHT STRANGE correlation would never have seen the light of day. The data that's been trickling in on [math]\varpi(x)[/math] has pretty much stopped,but this much, I'm reasonably sure of... At [math]\varpi(10^{12})[/math], the random fluctuations are still only about [math]\pm900[/math] or so, so at [math]x=10^{12}[/math], and [math]k=2*e[/math], we typically get calculations of [math]\alpha[/math] that are good to about 9 or 10 decimal places! Thus, we are on the right path to the most interesting "counting function" this side of [math]Li(x)[/math],and I can't think of a more worthwhile challenge for any "coder" than to go after [math]\varpi(10^{18})[/math], since that will probably allow us to determine the actual value of [math]k[/math] and the actual value of [math]\alpha[/math] to 14 decimal places or so! Since I know nothing about computers, all this is out of my hands, and all I can do is offer moral support and cheer the rest of you Hypographers on. :cheer: :cheer: :cheer: Yeah... I do feel pretty silly jumping around like this, but it's all in the name of science! So if i'm gonna be the cheerleader, why don't you be the project manager? Don. Quote
Rade Posted April 18, 2010 Report Posted April 18, 2010 Let me offer a very simplistic and non mathematical suggestion. I am not a physicist, I just enjoy thinking about things in new ways. As I understand the history of this problem with fine structure constant (reading Roger Penrose, The Road To Reality), there was time when the number 137.0359...was thought to be the exact number 137. So, let me take that approach to the problem. What would the number 137 mean for the atomic nucleus, in terms of a "coupling constant" for "fine structure" ? What does fine structure mean--structure of what ? As explained by Penrose, 1/137 is a type of coupling constant related to the nuclear electromagnetic interaction within the "structure" of the atomic nucleus, whereas the weak force coupling constant ~ 10^-6, and strong force = 1. So, we can place 1/137 within the fine structure of the atomic nucleus. If this is not correct, someone please correct me. OK, if this makes sense, then, what aspect of atomic nucleus would the number 137 be associated with ? Again, keeping in mind it is a coupling constant--that it is somehow related to how the atomic nucleus maintains its "structure" or identity as a thing that exists (I bring in a bit of philosophy, you do not have to agree it "really" exists--consider it the wavefunction if you wish). Penrose gives that 1/sq.rt 137 = charge on the PROTON, e = 0.0854246. So, clearly the number 137 has something to do with PROTONS. Penrose also explains that Sir Arthur Eddington spent a good part of his life trying to relate the number 137 (fine structure constant) to the total number of PROTONS in the universe. Here is my suggestion. Eddington was close to the solution, however 137 is not related to the total number of protons in the universe, but it is related to the Z value (= number of protons) for the largest possible isotope of elements that can have any fine "structure" via the interactions of all the fundamental forces (strong, weak, electromagnetic, gravity). I believe I read that recently an unstable isotope of a new element with Z = 117 was produced. I suggest that the number 137 is related to "charge on the proton" that places a "limit" on how many PROTONS can coexist to form an isotope, stable or unstable. That is, laws of nature will never allow for a Z = 138 element to exist. OK, I know this is simplistic, but if anyone would have a comment to show where my logic is false it would be appreciated. Quote
Don Blazys Posted April 19, 2010 Author Report Posted April 19, 2010 Let [math]\alpha=137.035999084^{-1}[/math], [math]\mu=1836.1526724718[/math], [math]\pi=3.1415926535898[/math] and [math]e=2.718281828459[/math], and let [math]\varpi(x)[/math] represent how many "polygonal numbers of rank greater than 2" there actually are under a given number [math]x[/math]. Then, for values of [math]k[/math] ranging from: [math]k=e^{\frac{\pi}{2}}=4.8104773809654...[/math] to [math]k=2*e=5.436563656918...[/math], the function: [math]\varpi(x)\[/math] ~ [math]B(x)-B(x)*\alpha*(\mu-k)^-1=[/math] [math]\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)-[/math] [math]\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)*\alpha*(\mu-k)^{-1}[/math] is not just "accurate", but "breathtakingly close" and "often perfect" for values of [math]x[/math] well beyond [math] x=10^{12}[/math]. However, if we change the above value of [math]\alpha[/math] to [math]137^{-1}[/math],then this function becomes much less accurate. Don. Quote
modest Posted April 19, 2010 Report Posted April 19, 2010 What does fine structure mean--structure of what ? 'Fine structure' refers to the spectral lines of an element (typically hydrogen). A spectral line, if you look closely enough (at its fine structure), is not a line. There's a fair Hyperphysics article: Hydrogen Fine Structure It refers, essentially, to the size of the 'splitting' of the spectral lines of hydrogen. OK, if this makes sense, then, what aspect of atomic nucleus would the number 137 be associated with ? A coupling constant determines the strength of an interaction. The fine structure constant determines the strength of the electromagnetic force on an electron. If it were different then the strength of that interaction would be different. I would presume there would be many other consequences because many parameters rely on the fine structure constant (as, for example, in the standard model). Penrose also explains that Sir Arthur Eddington spent a good part of his life trying to relate the number 137 (fine structure constant) to the total number of PROTONS in the universe. In fact, Eddington made many 'proofs' that the fine structure constant was exactly 1/136. When experimental measurement showed the value closer to 1/137 he made many more 'proofs' that it must be exactly 1/137. That is somewhat telling. Certainly, if you asked most physicists they would tell you that the fine structure constant is one of those dimensionless parameters that can't be mathematically derived (though many have tried and as this thread attests, still try) because it is a fundamental physical constant and not a mathematical constant. But, conventional wisdom is no reason not to try :naughty: ~modest Quote
Pyrotex Posted April 19, 2010 Report Posted April 19, 2010 I'm just curious, but why did you not simplify your equation to this: [math]\varpi(x)\[/math] ~ [math]B(x)-B(x)*\alpha*(\mu-k)^-1=[/math] [math]B(x)*(1-\alpha*(\mu-k)^-1)=[/math] [math]\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)*\left(1-\frac{\alpha}{(\mu-k)}\right)[/math] or even to: [math]\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)*\left(\frac{\mu-k-\alpha}{\mu-k}\right)[/math] Quote
Don Blazys Posted April 20, 2010 Author Report Posted April 20, 2010 To: Modest, Quoting Modest:In fact, Eddington made many 'proofs' that the fine structure constant was exactly 1/136. When experimental measurement showed the value closer to 1/137 he made many more 'proofs' that it must be exactly 1/137. That is somewhat telling. Certainly, if you asked most physicists they would tell you that the fine structure constant is one of those dimensionless parameters that can't be mathematically derived (though many have tried and as this thread attests, still try) because it is a fundamental physical constant and not a mathematical constant. But, conventional wisdom is no reason not to try Feynman suggested that it might be possible to derive the fine structure constantfrom some purely mathematical relationship, possibly involving [math]\pi[/math] or [math]e[/math]. Eddington, Gilson, and a host of others tried very hard to do just that. We, on the other hand, wanted nothing more than to provide Turtle with a "counting function" for his "non-figurate numbers"! The fine structure constant couldn't have been further from our minds. Little indeed did we know that such an innocent search for a "counting function" would result in this incredibly wierd... THING (as Capt. Kirk might say !). In short, this relationship between "Turtles numbers" and the fine structure constant was totally unexpected and came as a complete surpriseand even as somewhat of a shock :eek::eek2: ...and in that sense, was a completely accidental and serendipidous discovery. But even if the fine structure constant were not involved,this function would still be important, because it's the only counting function for "non-trivial polygonal numbers" that we have... and considering how extraordinarily difficult it is to calculate them by computer, this is something that we as Hypographers should be proud to contribute to the annals of mathematical literature. Don. Quote
Don Blazys Posted April 20, 2010 Author Report Posted April 20, 2010 To: Pyrotex, Quoting Pyrotex: I'm just curious, but why did you not simplify your equation to this: I wanted to show that we are essentially subtracting an "error term". Also, (and please keep this a secret) I tend to make really stupid blunders when simplifying,which I can't afford to do, since I already have a reputation as a "crank" and a "crackpot". Anyway, thanks for simplifying it for me. It looks beautiful ! Don. Quote
Don Blazys Posted April 20, 2010 Author Report Posted April 20, 2010 To: Pyrotex, I just remembered...:doh: You are the N.A.S.A. scientist! :) Wow! ...and Please... If you can find some spare time, can you and your colleagues use one of your more powerful computers to calculate [math]\varpi(x)[/math] to some really impressive value of [math]x[/math] as a "side project"? Don. P.S.I live in LaCrescenta, just a "stone throw" away from N.A.S.A./J.P.L in Pasadena,and my son used to play baseball for P.C.C. Quote
modest Posted April 20, 2010 Report Posted April 20, 2010 Let [math]\alpha=137.035999084^{-1}[/math], [math]\mu=1836.1526724718[/math], [math]\pi=3.1415926535898[/math] and [math]e=2.718281828459[/math], and let [math]\varpi(x)[/math] represent how many "polygonal numbers of rank greater than 2" there actually are under a given number [math]x[/math]. Then, for values of [math]k[/math] ranging from: [math]k=e^{\frac{\pi}{2}}=4.8104773809654...[/math] to [math]k=2*e=5.436563656918...[/math], the function: [math]\varpi(x)\[/math] ~ [math]B(x)-B(x)*\alpha*(\mu-k)^-1=[/math] [math]\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)-[/math] [math]\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)*\alpha*(\mu-k)^{-1}[/math] is not just "accurate", but "breathtakingly close" and "often perfect" for values of [math]x[/math] well beyond [math] x=10^{12}[/math]. Here is output I get for K = 4.8104773809654... and k = 5.436563656918... x = 1000 w(x,k) between: 627.71007436564630089122291274484991368745775233745 627.71007351024513457965178477714116184532220891766 x = 2000 w(x,k) between: 1262.8319109844068048188315028082273223220408149266 1262.8319092635042199699382106237052357330017876901 x = 3000 w(x,k) between: 1899.1731653019497577233789712653718430130891941325 1899.1731627138840137273547424210038181160456364800 x = 4000 w(x,k) between: 2536.1456366682526841060733678990588046556714731561 2536.1456332121636007272406283339013719508493169182 x = 5000 w(x,k) between: 3173.5214905136186968352994675234152455860152810070 3173.5214861889565715005253535614115241913733150994 x = 6000 w(x,k) between: 3811.1838793999144923299121008212487022788435608051 3811.1838742062888543136022104660349911590603128445 x = 7000 w(x,k) between: 4449.0633905891958270956782645636886500015399292956 4449.0633845263107966935060789212318882849924672414 x = 8000 w(x,k) between: 5087.1147978427337742849180904351725992055935668155 5087.1147909103551030757505388066485679864133735458 x = 9000 w(x,k) between: 5725.3066869078191496445513654626266729046411624560 5725.3066791057553984427665306702806303165813240016 x = 10000 w(x,k) between: 6363.6161852582677540104780344619532545008406233396 6363.6161765863586527742569376131926679173167096429 x = 11000 w(x,k) between: 7002.0260259843458586823403269283054852015736776984 7002.0260164424546675557380243860749699782828475886 x = 12000 w(x,k) between: 7640.5227944147797624176405697588206178684462937088 7640.5227840027880221585206611808647810127059702088 x = 13000 w(x,k) between: 8279.0958225657188960328414256860179558988657522187 8279.0958112835226850115746731703100369784604159804 x = 14000 w(x,k) between: 8917.7364608768676008081803480143852305588163358367 8917.7364487243747844354721233583182160727156697147 x = 15000 w(x,k) between: 9556.4375808703607296340964922209375570900127286821 9556.4375678474888875319240434969877222284610081661 x = 16000 w(x,k) between: 10195.193225084345697506656905435553518434366820237 10195.193211191020527726229491786278936942518230168 x = 17000 w(x,k) between: 10833.998354245635334437776992402841346714329201774 10833.998339481789402213638510477183303282603609887 x = 18000 w(x,k) between: 11472.848660574980908398259762880835830650658255667 11472.848644940552649317436984548829996760234757567 x = 19000 w(x,k) between: 12111.740427236359597984801623601750946253438883831 12111.740410731292512689031687340149415458002271113 x = 20000 w(x,k) between: 12750.670420712037805288736647093625377575810404421 12750.670403336279800840736001020918289683771125614 x = 21000 w(x,k) between: 13389.635807139828404914441557531301911421554997988 13389.635788893331250174146461903922396810761356316 x = 22000 w(x,k) between: 14028.634086397590677042683204910369707789178739808 14028.634067280309547868710408516885120554138463198 x = 23000 w(x,k) between: 14667.663039539710267965840542879606793788184838634 14667.663019551603364046985748200036589297529503939 x = 24000 w(x,k) between: 15306.720686421589478601589456098651268242222932498 15306.720665562617698034826773609784271879350020186 x = 25000 w(x,k) between: 15945.805251197832327692389987817839341244848446500 15945.805229467958988577629511681896291828660035417 x = 26000 w(x,k) between: 16584.915133976589557872722287424708600097606688996 16584.915111375780158535124136922210451180801621841 x = 27000 w(x,k) between: 17224.048887338490014082784795480346324566071298729 17224.048863866712025293497756279582888689981001186 x = 28000 w(x,k) between: 17863.205196737112230456749325992890140968371637960 17863.205172394334914362571569299797114391419228063 x = 29000 w(x,k) between: 18502.382864024440433047570352757287385735075082746 18502.382838810634684551013175257983771426637438465 x = 30000 w(x,k) between: 19141.580793513081528904407546795923314703672441691 19141.580767428219736037234490756809065500662775883 x = 31000 w(x,k) between: 19780.797980113560889582780220311356412396031180985 19780.797953157616810045243442762517041807502904840 x = 32000 w(x,k) between: 20420.033499181148207158856520145217016657234881482 20420.033471354096858694997547850249522054465939755 x = 33000 w(x,k) between: 21059.286497780415499363007066151517335515757133173 21059.286469082233062040031701700753686478926112857 x = 34000 w(x,k) between: 21698.556187132826466330671940799134407131778292698 21698.556157563490195110243582196464009567402727527 x = 35000 w(x,k) between: 22337.841836057237701464620238345688296505852242667 22337.841805616725847712997967524971076774784129070 x = 36000 w(x,k) between: 22977.142765248279040201750612609484141336086041244 22977.142733936570780996966590357132694975006739749 x = 37000 w(x,k) between: 23616.458342265401190655434563372087604321885042478 23616.458310082476564923061048865777091038824893879 x = 38000 w(x,k) between: 24255.787977127593337868919399947541112601679049441 24255.787944073433188521058998887575103444331868081 x = 39000 w(x,k) between: 24895.131118426628641325102703738359449723198731757 24895.131084501214562695986828410017326160947337514 x = 40000 w(x,k) between: 25534.487249886136331257231493017496281809240931491 25534.487215089450621227884463022575818436682410114 x = 41000 w(x,k) between: 26173.855887305547110008225371993243664182832132534 26173.855851637572726295714642790522382053451455487 x = 42000 w(x,k) between: 26813.236575837569091350450836894722109527550019341 26813.236539298289611507210320875500680602416945063 x = 43000 w(x,k) between: 27452.628887555754907466659201578062926564284359228 27452.628850145154492194568905209859546724654319131 x = 44000 w(x,k) between: 28092.032419275252622924583620359559720491463008690 28092.032380993315982358783695347959470818819585089 x = 45000 w(x,k) between: 28731.446790595257325360311538710630395969385370240 28731.446751441969688020631942378520296477193431542 x = 46000 w(x,k) between: 29370.871642136204938666824901315767871184044443313 29370.871602111552022811449033717200655410878791870 x = 47000 w(x,k) between: 30010.306633948540351274994625581397958686322602729 30010.306593052508338422546980386669342389273590420 x = 48000 w(x,k) between: 30649.751444073080512718811648429948963105964209192 30649.751402305656023132286027717158141147870729411 x = 49000 w(x,k) between: 31289.205767235685835034738779810487918708079604468 31289.205724596855904984240727811988146381558343164 x = 50000 w(x,k) between: 31928.669313661235670566708362228316157225323444258 31928.669270150987731208432228710907143869106920108 x = 10 w(x,k) between: 5.1383604794014263845590518660638373911117566520885 5.1383604723992135374080108326312590128230004330355 x = 100 w(x,k) between: 60.035148954194738355415409361983785673743876517396 60.035148872382866071854017310490695074639466376989 x = 1000 w(x,k) between: 627.71007436564630089122291274484991368745775233745 627.71007351024513457965178477714116184532220891766 x = 10000 w(x,k) between: 6363.6161852582677540104780344619532545008406233396 6363.6161765863586527742569376131926679173167096429 x = 100000 w(x,k) between: 63909.747700819666713475468535869653114508596105030 63909.747613727751266353685492854268790162442547906 x = 1000000 w(x,k) between: 639962.63142421471458573717442883107232134859204996 639962.63055211658573331424582696049839626537168373 x = 10000000 w(x,k) between: 6402362.1727245170375910786262008119289084878192159 6402362.1639998075047192536194368284050725464713521 x = 100000000 w(x,k) between: 64032273.271405350850420611152709464700847504502156 64032273.184146465777890310103352462155671874175567 x = 1000000000 w(x,k) between: 640350091.29887720742154318034621965906542506400872 640350090.42625107425276421264519685576781766930082 x = 10000000000 w(x,k) between: 6403587428.4303738789605300523692100447718769031872 6403587419.7039946498346198655418103387276407876287 x = 100000000000 w(x,k) between: 64036147870.151975778778671211883350568288269221743 64036147782.887810663084809771534825729387397151738 x = 1000000000000 w(x,k) between: 640362343855.93577583523769460790364022405895484844 640362342983.29294570391787510624667510436861246359 Is this the right post from Donk for a comparison? Ok, Don - here are your figures: NO FIG NONFIG PRIME* NONFIG/NONPRIME 1000 : 627 373 165 208 2000 : 1263 737 300 437 3000 : 1901 1099 427 672 4000 : 2540 1460 547 913 5000 : 3174 1826 666 1160 6000 : 3814 2186 780 1406 7000 : 4451 2549 897 1652 8000 : 5089 2911 1004 1907 9000 : 5723 3277 1114 2163 10000 : 6362 3638 1226 2412 11000 : 7000 4000 1332 2668 12000 : 7638 4362 1435 2927 13000 : 8273 4727 1544 3183 14000 : 8914 5086 1649 3437 15000 : 9555 5445 1751 3694 16000 : 10190 5810 1859 3951 17000 : 10833 6167 1957 4210 18000 : 11474 6526 2061 4465 19000 : 12112 6888 2155 4733 20000 : 12751 7249 2259 4990 21000 : 13390 7610 2357 5253 22000 : 14030 7970 2461 5509 23000 : 14664 8336 2561 5775 24000 : 15305 8695 2665 6030 25000 : 15942 9058 2759 6299 26000 : 16580 9420 2857 6563 27000 : 17216 9784 2958 6826 28000 : 17857 10143 3052 7091 29000 : 18495 10505 3150 7355 30000 : 19137 10863 3242 7621 31000 : 19777 11223 3337 7886 32000 : 20414 11586 3429 8157 33000 : 21055 11945 3535 8410 34000 : 21693 12307 3635 8672 35000 : 22338 12662 3729 8933 36000 : 22976 13024 3821 9203 37000 : 23614 13386 3920 9466 38000 : 24254 13746 4014 9732 39000 : 24891 14109 4104 10005 40000 : 25530 14470 4200 10270 41000 : 26173 14827 4288 10539 42000 : 26813 15187 4389 10798 43000 : 27454 15546 4491 11055 44000 : 28090 15910 4576 11334 45000 : 28733 16267 4672 11595 46000 : 29369 16631 4758 11873 47000 : 30009 16991 4848 12143 48000 : 30648 17352 4943 12409 49000 : 31287 17713 5032 12681 50000 : 31924 18076 5130 12946 [font="Courier New"] NO FIG NONFIG PRIME* NONFIG/NONPRIME 10 : 8 2 1 1 100 : 62 38 22 16 1,000 : 627 373 165 208 10,000 : 6,362 3,638 1,226 2,412 100,000 : 63,894 36,106 9,589 26,517 1,000,000 : 639,951 360,049 78,495 281,554 10,000,000 : 6,402,330 3,597,670 664,576 2,933,094 100,000,000 : 64,032,126 35,967,874 5,761,452 30,206,422 1,000,000,000 : 640,349,984 359,650,016 50,847,531 308,802,485[/font]* Only nonfigurate primes - 2, 3 and 5 have been left out of the count. I now have a massive amount of raw data, with figurates running up to a billion, and can pull it out in just about any form you need. Would powers of 2 be better than powers of 10? You'd get 29 data points instead of 8. Or powers of e, for 20 data points? I've checked the data as best I can - one check is that my primes list matches that on wikipedia, remembering to subtract 3 from the powers-of-ten value. If anyone comes up with an alternative nonfig-generator, I'd love to send a million or so nonfigs to them as a crosscheck :) ~modest Quote
Don Blazys Posted April 21, 2010 Author Report Posted April 21, 2010 [math]x[/math]______[math]\varpi(x)[/math]_____[math]B(x)*\left(\frac{\mu-2*e-\alpha}{\mu-2*e}\right)[/math] ____Diff____%Error___10^1____3_______________5_____________________2_____.6666610^2____57______________60____________________3_____.0526310^3____622_____________628___________________6_____.0096410^4____6,357___________6,364__________________7_____.0011010^5____63,889__________63,910_________________21____.0003210^6____639,946_________639,963________________17____.0000265610^7____6,402,325________6,402,362______________37____.0000057791510^8____64,032,121_______64,032,273_____________152___.0000023894310^9____640,349,979______640,350,090____________111___.0000001733410^10___6,403,587,409____6,403,587,420___________11____.0000000134310^11___64,036,148,166___64,036,147,783__________-383__-.0000000059810^12___640,362,343,980__640,362,342,983_________-997__-.00000000156 [math]B(x)*\left(\frac{\mu-2*e-\alpha}{\mu-2*e}\right)[/math] crosses [math]\varpi(x)[/math] many, many times between [math]x=[/math]10^11, and [math]x=[/math]10^12 . Don. Quote
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