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# User:Jani Melik

## Basic personal data

Born: 19640222, Celje, Slovenia.

• 19640222-th prime is 366483151.
• factor set of 19640222 is {2, 7, 1402873}.

Education:

• Technical faculty (at that time), University of Maribor, 1990, Design and engineering construction course, diploma thesis Computer aided design of tools for sheet metal forming parts (mentor mag. Martin Prašnički, višji predavatelj TF);
• Faculty of Mechanical Engineering, University of Ljubljana 2009, Production mechanical engineering course, VSŠ diploma thesis Rough estimation of number of operations in sheet metal forming (mentor red. prof. dr. Karl Kuzman, univ. dipl. inž. str., COBISS 11050779).

Membership - societies:

User webpages:

## General characteristics

Mathematics is the queen of science, and number theory is the queen of mathematics.Carl Friedrich Gauss
• irationality
• primality
• divisibility
• smallness, finiteness, infiniteness (magnitude)
• reversion (simmetry)
• neighbourhood
• planarity
• periodicity
• rootness
• convexity

## NumbԐrs

Answԑrs are numbԑrs, and numbԑrs are answԑrs. And numbԑrs are also qѵԑstions.
What is tӀme to numbԑrs, as like numbԑrs are to mѧttԑr (or fѵnctions to spѧce)?

It is quite a while well known (is it?) that numbers, as also Alexander Adam wrote: "because the universe of (natural) numbers is similar to an enigma" and others said, are perhaps more then 'just' numbԑrs - let say similar to atoms, which composite matter. For instance well known number 23. Is it a Ramanujan prime or not? No, this one is not: A104272. But nevertheless, why in the end it should or it should not be? So, it is 6-th non-Ramanujan prime: A174635. Number theory probably does not ask questions like: "Why 23 is non-Ramanujan prime, and not Ramanujan one?", although Hardy's school is perhaps still much alive. But in the Balthazar's upside-down (inside-out) city of M (ver. U-DI-OCM) taxi cabs use car plates as simply ʍʇ32: A011541, and so professor Balthazar's taxi cab numbers are: 2, 9271, 91393578, 8429032743696, 96426967295688984, 44356021345218591335142, ... Of course, normally we would call them simply reversal taxi cab numbers, although prof. Balthazar was quite ingenuous old geek, and also this masterpiece of cartoon from Zagreb drawing school was unfortunately probably unknown outside the old ex-state.

It is also well known that 23 is (after number 2) the second prime which is not a member of twin primes (A001097, A007510), or the smallest odd non twin prime. Sometimes 'smaller' can be handy, if not for nothing, at least it can be easily remembered. Regarding remembering of numbers here's an interesting allegeable story. Famous Slovene mathematician France Križanič, who lectured mathematical calculus, once wrote ${\displaystyle e=2.718281828459045\ldots \,}$. This seemed fascinating to students who knew only as much digits as TI-30 could show. Križanič in his style was silenced a while, and afterwards he explained: "It is very simple. Two point seven," and shows, "you know all. The same year Goya died," and shows first 1828, "Tolstoy was born," and shows the other 1828. "And after that some short multiplication follows," and shows to 45 90 45. But what about the 42-nd twin prime or ${\displaystyle 2^{42}\equiv 4398046511104\,}$-th? Term a(42) can be easily find here to be 419. Some say, if an algorythm for computing the 4398046511104-th twin prime does not exits, there might be a 'computer' (think for example to one from Adams' The Hitchhiker's Guide to the Galaxy somewhere (DT or E), or let say in akashic space). If you have a black boxlike computer from Akasha, you would just type in (like in Maple's worksheet or in some OS's command line) 4398046511104 Өts0ps:tpѲ or something like that, and you would 'get' an answer instantly... In SF answԑrs are very often (again) numbԑrs - like 42.

 ${\displaystyle {\frac {1}{23}}={\frac {1}{24}}+{\frac {1}{552}}\!\,}$ 1 ${\displaystyle {\frac {1}{23}}={\frac {1}{24}}+{\frac {1}{553}}+{\frac {1}{305256}}\ldots \!\,}$ 87 ${\displaystyle {\frac {1}{23}}={\frac {1}{24}}+{\frac {1}{553}}+{\frac {1}{305257}}+{\frac {1}{93181530792}}\ldots \!\,}$ 12326 ${\displaystyle {\frac {2}{23}}={\frac {1}{12}}+{\frac {1}{276}}\!\,}$ 1 ${\displaystyle {\frac {2}{23}}={\frac {1}{12}}+{\frac {1}{277}}+{\frac {1}{76452}}\ldots \!\,}$ 39 ${\displaystyle {\frac {2}{23}}={\frac {1}{12}}+{\frac {1}{277}}+{\frac {1}{76453}}+{\frac {1}{5844984756}}\ldots \!\,}$ 4203 ${\displaystyle {\frac {3}{23}}={\frac {1}{8}}+{\frac {1}{184}}\!\,}$ 1 ${\displaystyle {\frac {3}{23}}={\frac {1}{8}}+{\frac {1}{185}}+{\frac {1}{34040}}\ldots \!\,}$ 20 ${\displaystyle {\frac {3}{23}}={\frac {1}{8}}+{\frac {1}{185}}+{\frac {1}{34041}}+{\frac {1}{1158755640}}\ldots \!\,}$ 1970 ${\displaystyle {\frac {4}{23}}={\frac {1}{6}}+{\frac {1}{138}}\!\,}$ 1 ${\displaystyle {\frac {4}{23}}={\frac {1}{6}}+{\frac {1}{139}}+{\frac {1}{19182}}\ldots \!\,}$ 19 ${\displaystyle {\frac {4}{23}}={\frac {1}{6}}+{\frac {1}{139}}+{\frac {1}{19183}}+{\frac {1}{367968306}}\ldots \!\,}$ 1396 ${\displaystyle {\frac {5}{23}}\!\,}$ 0 ${\displaystyle {\frac {5}{23}}={\frac {1}{5}}+{\frac {1}{58}}+{\frac {1}{6670}}\ldots \!\,}$ 8 ${\displaystyle {\frac {5}{23}}={\frac {1}{5}}+{\frac {1}{58}}+{\frac {1}{6671}}+{\frac {1}{44495570}}\ldots \!\,}$ 467 ${\displaystyle {\frac {6}{23}}={\frac {1}{4}}+{\frac {1}{92}}\!\,}$ 1 ${\displaystyle {\frac {6}{23}}={\frac {1}{4}}+{\frac {1}{93}}+{\frac {1}{8556}}\ldots \!\,}$ 9 ${\displaystyle {\frac {6}{23}}={\frac {1}{4}}+{\frac {1}{93}}+{\frac {1}{8557}}+{\frac {1}{73213692}}\ldots \!\,}$ 636 ${\displaystyle {\frac {7}{23}}\!\,}$ 0 ${\displaystyle {\frac {7}{23}}={\frac {1}{4}}+{\frac {1}{20}}+{\frac {1}{230}}\ldots \!\,}$ 3 ${\displaystyle {\frac {7}{23}}={\frac {1}{4}}+{\frac {1}{19}}+{\frac {1}{583}}+{\frac {1}{1019084}}\ldots \!\,}$ 97
• 30 42 102 105 138 154 165 170 186 195 231 246 266 282 370 399 426 ...
• 4 9 12 18 45 50 56 60 64 72 76 81 99 108 120 144 150 160 176 180 192 198 225 228 236 240 ...
• 6 15 21 26 34 39 69 86 93 111 129 134 205 217 254 ...
Full search for last two sequences find both 16×16 magic squares: Franklin's (A124472) and Stifel's (A203813). Strange (or not) or what?
• Number of solutions of Diophantine equation:
${\displaystyle {\frac {3}{n}}={\frac {1}{x}}+{\frac {1}{y}};\qquad 00;\quad n,x,y\in \mathbb {N} ^{+}\!\,}$
with Egyptian fractions in mini Erdős–Straus conjecture:
0 1 0 1 1 1 0 2 1 2 1 2 0 3 1 2 1 4 0 4 1 2 1 3 1 3 2 3 1 4 0 3 1 2 3 7 0 3 1 5 1 4 0 4 4 2 1 4 0 4 1 3 1 7 2 6 1 2 1 7 0 3 4 3 3 4 0 4 1 6 1 10 0 3 2 3 3 4 0 7 3 2 1 7 2 3 1 5 1 13 0 4 1 2 3 5 0 5 4 6 ...
Full search finds 19900 results. By 'full search' I mean my ignorance on how to search only in data fields.
• Neighbours of semiprimes (both side) (A001358) (to be double checked):
3 5 8 9 10 11 13 14 15 16 20 21 22 23 24 25 26 27 32 33 34 35 36 37 38 39 40 45 47 48 50 52 54 56 57 58 59 61 63 64 66 68 70 73 75 76 78 81 83 84 85 86 87 88 90 92 93 94 95 96 105 107 ...
Full search also finds A124472 and A203813 in data field. Is this more common situation?
• Number of edges in planar n-dimensional hypercube (sub)graphs for ${\displaystyle 0\leq n\leq 10\,}$:
 ${\displaystyle n\in \mathbb {N} _{0}\,}$ 0 1 2 3 4 5 6 7 8 9 10 ... A001477 0 1 4 12 32 80 192 448 1024 2304 5120 ... A001787 0 1 4 12 24 41 77 138 270 529 1041 ...
 ${\displaystyle n\in \mathbb {N} ^{+}\,}$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 ... A000027 ${\displaystyle p_{i}+\{1\}\,}$ 1 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 ... A008578 ${\displaystyle \lfloor p_{i}/n\rfloor \,}$ 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 ... A171622 ${\displaystyle p_{i}\mod i+1\,}$ 0 0 0 1 2 5 6 1 1 3 7 7 11 13 13 15 2 5 4 7 8 7 10 11 14 19 20 ... A004650
 ${\displaystyle n\in \mathbb {N} _{0}\,}$ (-1) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 #-- - 3 3 7 8 7 4 5 9 9 / / / / / (3) 1 5 9 9 8 6 1 7 2 9 0 9 2 5 A055199 ${\displaystyle |\ominus |\,}$ - 2 2 2 1 1 2 4 2 7 / / / / / ${\displaystyle |\oplus -10|\,}$ - 4 8 6 7 5 0 6 6 1 / / / / /
 ${\displaystyle n\in \mathbb {N} _{0}\,}$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... #-- ${\displaystyle \pi (10^{n})\,}$ (0) 4 25 168 1229 9592 78498 664579 5761455 50847534 455052511 4118054813 37607912018 346065536839 3204941750802 29844570422669 ... A006880 ${\displaystyle \pi _{\rm {G}}(10^{n})\,}$ (0) 2 13 87 619 4808 39322 332398 2880950 25424042 227529235 2059034532 18803987677 173032827655 1602470967129 14922285687184 ... A091099 ${\displaystyle \pi (10^{n})-\pi _{\rm {G}}(10^{n})\,}$ (0) 2 12 81 610 4784 39176 332181 2880505 25423492 227523276 2059020281 18803924341 173032709184 1602470783673 14922284735485 ... ${\displaystyle 2\pi _{\rm {G}}(10^{n})-\pi (10^{n})\,}$ (0) 0 1 6 9 24 146 217 445 550 5959 14251 63336 118471 183456 951699 ...
• 1 1 2 5 6 7 5 4 9 6 1 6 4 1 7 5 5 9 4 5 1 6 6 1 6 4 1 ...
• 2 5 4 1 5 5
• 1 6 4 4 6 6 4 8 8 7 9 9 6 3 9 6 1 9 6 9 9 7 6 6 9 6 6 6 3 9 8 6 6 9 3 3 9 3 9 9 9 ...
• 1 1 1 1 1 1 6 1 1 1 2 1 6 6 1 1 7 1 9 1 6 2 4 1 1 6 3 6 1 1 6 1 2 7 6 1 3 9 6 1 5 6 3 2 1 4 1 1 6 1 7 6 4 3 2 6 9 1 4 1 6 6 6 1 6 2 6 7 4 6 8 1 8 3 1 9 6 6 4 1 9 5 5 6 7 3 1 2 8 1 ...
• 3 7 6 1 4 1 1 1 2 1 3 1 5 2 1 1 2 2 2 2 1 3 2 1 1 6 3 4 1 4 2 6 6 9 1 2 2 6 3 5 1 1 6 8 1 7 1 2 3 7 1 2 1 1 3 1 1 1 3 1 1 8 1 1 2 1 6 1 1 5 2 2 3 1 2 4 4 7 1 8 9 1 4 1 2 2 1 4 1 2 6 1 2 1 3 1 2 1 ...
• 1 7 7 5 9 5 5 1 7 8 4 3 1 5 6 2 1 4 9 4 4 7 9 7 7 4 ...
• 3 4 9 4 7 2 9 2 4 6 4 1 9 1 1 2 5 3 2 7 9 7 5 3 8 ...
• 9 9 9 9 1 5 6 8 7 9 3 6 9 4 2 6 2 4 9 6 3 9 7 8 6 5 1 9 9 9 9 1 5 6 8 7 9 3 6 9 4 2 6 2 4 9 6 3 9 7 8 6 5 1 9 9 9 9 1 5 6 8 7 9 3 6 9 4 2 6 2 4 9 6 3 9 7 8 6 5 1 9 9 9 9 1 5 6 8 7 9 3 6 9 4 2 6 2 4 9 6 ...
• 9 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 ...

### Lθgs

 ${\displaystyle \operatorname {lb} \,42\,}$ 5.3923174227787602888957082611796473174008410336586218441330443786114190766565515490201414740882990271 ... [5; 2, 1, 1, 4, 1, 1, 1, 1, 5, 2, 8, 1, 7, 1, 10, 2, 1, ...] ${\displaystyle \log _{3}42\,}$ 3.4021735027328796971674554214252185723660569747261239072396475211185714000837270158954736778869607218 ... [3; 2, 2, 18, 287, 1, 8, 3, 1, 4, 1, 3, 1, 2, 1, 9, 39, 2, 1, 2, ...] ${\displaystyle \log _{5}42\,}$ 2.3223447076815459025679892735823779562110172616637607876882689303072864464622987700196696393392623891 ... [2; 3, 9, 1, 3, 1, 1, 34, 3, 1, 3, 1, 5, 5, 1, 2, 1, 1, 2, 5, 1, 2, ...] ${\displaystyle \log _{18}42\,}$ 1.2931449416748841651796503304996953819299473677816938079981047064680830296890449755399151165006098272 ... [1; 3, 2, 2, 3, 6, 1, 6, 1, 3, 12, 10, 1, 4, 1, 1, 1, 1, 8, 3, 7, ...] ${\displaystyle \log _{23}42\,}$ 1.1920511922155704915707878897888844984631928094436115325615969431904119406383487244100337567415074444 ... [1; 5, 4, 1, 4, 1, 23, 1, 2, 2, 1, 3, 1, 2, 1, 3, 3, 1, 1, 2, 1, 2, ...] ${\displaystyle \log _{41}42\,}$ 1.0064890491274204382468417950335949430615664633988939892971150146932660472161695672404105311456784088 ... [1; 154, 9, 2, 4, 1, 5, 1, 1, 1, 13, 5, 1, 1, 42, 2, 2, 2, 6, 1, ... ]