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User:Bill McEachen

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The email shown in sequences is obsolete...use

married, 2 kids, control systems engineer, interest in prime numbers, born 1962 live in southwest Virginia USA, BS,MSEE,PE
I will post brief info on the few sequences I have authored on the discussion page here shortly if not already


My Sequences

(I'm lax, I sometimes say integers for whole numbers ...)
A129912 June 2007 relates to my main PPP conjecture (primes as f{primorials,primorial products}. This is effectively my mild claim to fame, in that it works to explain the way that the primes are distributed. For clarity I will repeat my original conjecture here:

   Every prime number >2 must have an absolute distance to a sequence entry
(primorials, primorial products) that is itself prime, aside from the special
cases prime=2 and those primes immediately adjacent to a sequence entry
(primorials, primorial products). The property is required but not sufficient considers distances no larger than the candidate

Rephrasing, this merely means every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product. (the distance will be a prime smaller than the candidate)

The usefulness of the property is evident from the Wikicommons image referenced in the sequence. The normalization is merely -1 to +1, where 0 indicates the closest PPP entry was very close, and +/- 1 indicates the closest PPP entry was as far away as possible. Note that the generation of the entries for A129912 appears related to A071562 in a slightly convoluted way. My website details more about the conjecture, including other independent work by Sokol and Potter. Note OEIS begins the sequence at 1, while I begin it at 2 (this seems to happen a lot, I have given up trying to make sense of it).

See comments on the talk page of the Primorial article here, where a poster says more that appropriate wording is that the offset must be coprime to all of the governing factors of the PPP entries.--Bill McEachen 21:39, 25 December 2010 (UTC).

Also, as a clarification, I cannot compromise in the logic about the 1st term should start at 2. By definition, the sequence treats prime factorials, which do not exist below the first prime, which is 2.

A117825 May 2006 relates to distance from the Highly Composite Numbers and the nearest prime. This relates to Goldbach's Conjecture, my HCN conjecture (see my website), and Fortune's conjecture. This has an interesting pinplot. I posted an image based on CRG data for first 20K here:svg image. Also mentioned in A228945 & A228943.

A147517 Nov 2008 relates to the symmetrical primes as a f{A002110, primorials}. Here symmetrical primes means the pair sum to twice the primorial center.

A077287 Aug 2003 concerns a mechanism that generates a lot of prime numbers with very volatile numeric spread (see the OEIS scatterplot). I derived while looking for good prime generators. Possible rifo A023057

A113972 Feb 2006 is a form I derived that yields an interesting sequence of primes, with many hi/lo transitions. The OEIS scatterplot here depicts it:[1]

A156053 Feb 2009 shows the Twin Primes (lower of pair) encountered by the Bernoulli Number denominators. It excludes duplicates, and covers through first the 15000 BN. I have an image posted here [2]. From Mr Cloitre the BN denom are produced via a(n)=prod(p=2, 2*n+1, if(isprime(p), if((2*n)%(p-1), 1, p), 1)) . Mr Karrhus and his new TP work (Mar 2013) is a bit similar, but he produces larger Twin Primes. He produces 27 TPs evaluating thru Prime ~ 14000. I will add its plot here shortly

A172069 Jan 2010 merely relates the adjacency of primes to entries of A129912. The OEIS plots here depict it:[3]. I must analyze its mean (0.5?) and (5) pair sum ratios later (skewed to the lower half).

A147853 Nov 2008 is a spinoff of A147517 and relates to Goldbach partitions of 2*primorials (A002110), providing a very good spread of Goldbach pairs

A102648 Feb 2005 is just a nice function to generate an interesting spread of values (see OEIS pinplot). It is a simple mechanism to quickly generate prime numbers from a fixed range of small integer seeds in a fairly arbitrary way. The submitted version generates values <=20 but this range is dependent on the gain (100) used. The OEIS plots here depict it: [4]. I have a rejected sequence that does it better.

A156703 Feb 2009 is a nice infinite, diverging sequence that is pretty volatile term-by-term. It is formed from ratios of the whole numbers. All but the 3rd entry appear to contain the digit 5 or 9. I posted an image here [5]. In Jan 2014 I tied this sequence to A158911, in turn connected to the Hamming Numbers. I also made a new conjecture about the sequence entries.

A005846 this relates to a classic eqn for generating primes. I have an image posted here [6]

A181616 produces volatile list of primes from a seed, with 2 distinct resulting limit lines for the log scatterplot.

A184938 this relates to bolt-tightening patterns but the reference was deleted when approved. That reference is 1

A218391 this relates to an observation by John Richardson on distinguishing composites from primes. On its face it would not help practically with primality testing, but I'm thinking.

A226095 this produces primes from the prime factorization of integers>1 (was composites). Good "multiplier" as far as size of primes ID'd vs base composite. Nice volatility, thanks to CRG once again for his code assist ...doubt it has any improved hit rate on encountering primes...thanks to TD Noe for his observations to fix my comments.

A228446 this relates to Sun's conjecture that every odd n>5 = p+x(x+1) where p is an odd prime, x an integer. Nice plot, generates some primes higher than largest pf of originating n. It has a beautiful patterned scatterplot.

I may submit a sequence based on Papalexis' data compression method (Jan 2014) (normalized R values as F(bit string length and which bits are set). I would like to submit a sequence w/ limited terms I worked up regarding maximal production of primes from quadratics.

I submitted a candidate sequence Jan 12 2014, a companion to my A156703. It produces highly volatile entries. It stems from repeating decimals of pairs of adjacent whole numbers. (tentative 235589). Entries seem to always contain a 9,8,or 6 (akin to A156703 entries having 9 or 5).

A240563. Its form can generate families of nicely spread prime numbers via concatenation.

I submitted sequence A242902 which generates primes from sets of twin prime sets. It has a "hit" rate of ~ 30% for the first 10K evaluations (prime range 3-2500000). One can compare this to primepi of 40% thru 10K (smaller primes than the former however).

I submitted draft sequence A245694 which stems from a MMF post/conjecture. It may be a short sequence but seems to consist of prime squares.

I would delete my seq A087867 as useless.


Thanks to direction from CRG, I was able to provide the solution set to the generalized conjecture mentioned by Winkler in A051451. In a nutshell, the majority of solutions are the multiples of 6. The rest stem from A053319. The generalized conjecture is: There exists a class of (even) numbers such that each member lies between two pairs of Twin primes (TP1 thru TP4 in increasing order). A governing relation is each member = (TP3-TP1). An example is member 6 and (5,7,11,13). Members of the class include all multiples of 6 as well as some others related to A053319, as described there. As the multiples of 6 are infinite, this implies the Twin Primes are as well, in a backhanded way, which most mathematicians believe.

Minor observations

A few other sequences that I've interacted with:

A005097 I made connection to Goldback partitions
A000079 I tied this to the divisors of the primorials (see lower)
A001359 I made connection to the generalized Winkler conjecture
A109852 I conjectured regarding every odd prime >4 being encountered in order
A079149 I noted the connection between entries and a sieving involving (a+b) and (a^2-b^2) for consecutive composites
A002110 I noted that the xth root of the xth primorial seems to have magnitude on the order of its # factors.
A002182 I noted a(n) often corresponds to P(n,m) = number of permutations of n things taken m at a time.
A158869 I asked whether the pattern of rightmost digits) (1,5,7,7) will continue?
A006562 I noted similarity of plot with A013916
A077287 Here is a plot for kissing number aligned to A129912 entries for correlation (I start the latter at entry=2): [7]
A000849 These entries highly correlate to A147517 as follows:
take adjacent differences of the A147517 entries, and plot the log of those values vs A000849.
The resulting linear fit is Y=Ax+B where Y is the A000849, X =A147517, (A,B)=(1.0451,0.241579).
The fit to the admittedly small dataset is better than 0.9999
For clarity, the X dataset is (5,24,160,1374,15511,...) and Y dataset (10,46,343,3248,42331,...)
The fit yields (9,48,351,3320,41804,...)


I added a comment about the frequency of primes to the sequence entries. Specifically, the "hit" rate is higher than average, after applying PrimePi with adjust for odd, non multiples of 5 as the baseline. I did the analysis in Gnumeric and can supply that data.--Bill McEachen 06:37, 6 December 2010 (UTC)

I also compared Primepi vs that derived from this sequence, the ratio of which seems to be tending towards 1.5
Specifically, for xample, entry 17 is 132. Primepi(132)=32. rat1=32/132/0.4=0.61
The sum of adjacent primes thru the 17th entry is 13 (primes adjacent to each entry). this value 13/entry ptr=17 is 0.76. I was trying to see how relevant a forecaster for PrimePi the sequence is. I looked at the first 1500 entries only and here is the ratio plot:[8]
--Bill McEachen 04:08, 23 January 2011 (UTC)


I continue to find new tie-ins to OEIS for my work. I make use of the prime sets named at [9], and they all appear in OEIS, where anciliary notes are often quite useful.--Bill McEachen 06:40, 6 December 2010 (UTC)

Radix transformations

I entered a sequence (not accepted) relating this subject with a specialized format. Considering unsigned integers only, the form is xxx.yyy where yyy is the remainder

10. Now unlike hexadecimal or other bases where we use more than 10 numeric characters (like A thru F in hex), we we constrain ourselves to digits 0-9. Then, the sequence, interest xpresses the # bases each number can be xpressed in. The first decrease is at 110, which begins the coverage gap for base 11 between 11^2 and 9*11+10. The interest to me is the local minima seen in the sequence. I can provide Python code to produce the representation for any base (fails if in coverage gap).
The sequence begins 1,1,2,2,2,2,3,3,4 considering only primes radix. The sequence was not accepted into OEIS.

Named prime sets

I have a list of named prime sets: User:Bill McEachen/Named prime sets and User:Bill McEachen/OEIS_prime_sets.

Mersenne Numbers

I've submitted a sequence A186283 related to A186253. Very volatile but well-bounded at lower end. I know of the prime power theorem, but will the factor hold for all powers? I also submitted the dual case (2^n+1) as A185343. I thank CRG and TD Noe for their assistance.

Sequences of interest

I like A166448 because of its graph and concept captured. It follows a 2nd-degree polynomial very well. Thru 500 terms I got Y=3.698x^2-231.7x+7484, R^2=0.9999+

I updated the related A166449 with a b-file if you're interested. Charles R Greathouse IV 07:13, 24 February 2011 (UTC)

As CRG et al point out, there is an established power-ln relation involved.

A084317 (from concatenation of prime factorizations) its bfile is short but I did note that the entries of A002110 seem to appear (+1). So, we have 2,6,30,210,2310,30030 and the seq contains entries 3,7,31,211,2311. Will there often be such a correspondence? One ref for prime factorizations is 1. The next would be 30031, which won't be encountered in A084317. The rest wouldn't make much sense either now that I look at them ...oh well

A229159 from M. Lagneau is very interesting (Sept 2013)

Since you find this interesting I looked at it as well. It seems to me that using the n-th prime is artificial, and I wonder if there might be another interesting sequence if you take this away and use numbers or odd numbers in place of primes. Perhaps even a sequence for "numbers n such that there is a triangle with leg n, all legs integers, and integral area". (Feel free to submit any of these if you feel so inclined and don't find them in the OEIS.) But I haven't had time to look at this in sufficient detail. Charles R Greathouse IV 20:38, 25 September 2013 (UTC)
I actually emailed Mr Lagneau giving what I think proves his conjecture (must be straightfwd for me to be able to do it)--Bill McEachen 00:37, 15 November 2013 (UTC)


This was one of those strange "hits" one gets from fiddling with numbers. I was working with bit strings, considering bit transitions, # adjacent bits set from left etc. I concluded:

the final bound is (N-1 left aligned bits ) + 2^(N -#set bits-1) - A084639(ptr) with ptr=#bit transitions-1

This takes a bit of xplanation but crudely, for a(32),input to Wolfram Alpha is:

111111111111111111111111111111110 base 2 - 101010101010101010101010101010101 base 2

this gives 2863311529. We have 32-bit transitions in the right portion. We have N=33, and there are 17 set bits. This gives 2^(33-17-1)=32768. A(32)=2863311529. So, final bound ~ 5726655829. It computes to 5726623061.
I only now see CRG's Pari code for the seq, so it transforms the above "A084639(ptr)" to 2^(ptr+2)/3-if(ptr%2, 5, 4)/3--Bill McEachen 01:46, 9 November 2013 (UTC)


My fit to the curve is 1.06766 + 1.02336*ln(x-0.574)...

FWIW there should be about 1.37*sqrt(x)/log(x) members of that sequence up to x. Put another way, a(n) should be about 0.132 n^2 log n. - Charles R Greathouse IV 07:45, 3 November 2013 (UTC)
I forgot I had fit log(Y), so the fit is log(Y)=1.06766 + 1.02336*ln(x-0.574), where x is ptr, Y=a(ptr). For example, actual is (4600,4694305229). This gives (4600,9.67). Fit gives 9.70...If I treat your eq'n above even using ln, it estimates 23556648, pretty shy of actual.--Bill McEachen 00:37, 15 November 2013 (UTC)

A248863 (proposed)

this is merely a bit pattern that results from concatenation from A011545 terms. This relates to Pi. I must graph it against A004601...--Bill McEachen 02:25, 4 March 2015 (UTC)


This goes back to Oct 2008. It begins from pf=7 1,2,4,8,16. length(divisors(7))=2 length(divisors(11*7))=4 [1,7,11,77] length(divisors(13*11*7))=8 [1, 7, 11, 13, 77, 91, 143, 1001] length(divisors(17*13*11*7))=16 power of 2 should match #terms in Prod(P), p from prime(4) to prime(n) SO, use prime(8) to prime(4). divisors should = 2^5=32 Prime(20) should yield 2^17. It does (131072). It's obviously thinking about it, but any basic connection between (powers of 2) and (products of primes) is nice.


some basic functions that are often useful (see refcard for all, or use ? xxx where xxx is command):

  • print, write
  • for, while, if
  • List(), listput,listsort,setsearch
  • sumdigits, digits
  • isprime,ispseudoprime,nextprime,precprime,prime(*)
  • x%y (mod), abs, ++ (increment), -- (decrement), max, min, floor, ceil, round, truncate
  • random, length, Str (convert to string),
  • divisors, sigma, Pi, prodeuler (product over primes), numerator, denominator
  • issquare, ispower, fibonacci, sumformal

Idea for page

I think a page covering "popular sequence topics" and/or phrases would be useful. There are obvious ones; a very quick stab might include (counts as of May 2014):

  • keyword: easy (~55000)
  • keyword: base (~30000)
  • expansion (~22000)
  • "Vincenzo Librandi" (~21600)
  • series (~18600)
  • "Clark Kimberling" (~17500)(author)
  • binomial (~ 14700)
  • binary (~11000)
  • "decimal expansion" (~10500)
  • "Reinhard Zumkeller" (~10500)(author)
  • "T. D. Noe" (~10500)(author)
  • "Robert G. Wilson" (~10000)(author)
  • divisors (~9700)
  • partitions (~9500)
  • polymonial (~9500)
  • permutations (~9000)
  • Greathouse (~ 9000)(author)
  • transform (~8600)
  • lattice (~7000)
  • Fibonacci (~ 7000)
  • roots (~6800)
  • matrix (~6200)
  • "Alois P. Heinz" (~6200)
  • compositions (~6200)
  • "Jean-Fran├žois Alcover" (~5700)(author)
  • "Amarnath Murthy" (~5500)(author)
  • phi (~5100)
  • keyword:fini (~5100)
  • keyword: hard (~4800)
  • "Colin Barker" (~4800)(author)
  • "Benoit Cloitre" (~4600)(author)
  • Euler (~4200)
  • "numerators of" (~4000)
  • sorting (~4000)
  • nodes (~3900)
  • "Vladeta Jovovic" (~3900)(author)
  • "David W. Wilson" (~3900)(author)
  • "Roger Bagula" (~3800)(author)
  • "prime number" (~3800)
  • sigma (~3800)
  • convergent (~3700)
  • trees (~3500)
  • concatenation (~3400)
  • "Jonathan Vos Post" (~3300)(author)
  • "M. F. Hasler" (~3300)(author)
  • "denominators of" (~3200)
  • "Wolfdieter Lang" (~3100)(author)
  • "Michel Marcus" (~3100)(author)
  • sin( (~3000)
  • modulo (~2950)
  • "Joerg Arndt" (~2900)
  • factorial (~2800)
  • palindrome (~2750)
  • derivative (~2600)
  • asymptote (~2500)
  • "Ralf Stephan" (~2500)(author)
  • "number of ways" (~2500)
  • "triangular numbers" (~ 2400)
  • "first difference" (~2400)
  • intersections (~2300)
  • "smallest prime" (~2300)
  • primepi (~2100)
  • Pascal (~2050)
  • Ramanujan (~2000)
  • semiprime (~2000)
  • Lucas (~2000)
  • theta (~1800)
  • "Juri-Stepan Gerasimov" (~1800)(author)
  • "Twin prime" (~1625)
  • tau (~1500)
  • "Peter Luschny" (~1500)(author)
  • "Michel Lagneau" (~1400)(author)
  • "Vladimir Shevelev" (~1350)(author)
  • gap (~1300)
  • symmetry (~1250)
  • "Lekraj Beedassy" (~1200)(author)
  • "Alonso del Arte" (~1200)(author)
  • progression (~1200)
  • Bernoulli (~1100)
  • "binary representation" (~1050)
  • Chebyshev (~1000)
  • Dirichlet (~1000)
  • Fermat (~950)
  • packing (~850)
  • irreducible (~850)
  • Mersenne (~800)
  • harmonic (~780)
  • "Zhi-Wei Sun" (~721)(author)
  • "Don Reble" (~721)(author)
  • "Jonathan Sondow" (~700)(author)
  • "Gary Detlefs" (~700)(author)
  • "Jud McCranie" (~700)(author)
  • "first occurrence" (~700)
  • "characteristic polynomial" (~700)
  • Fourier (~700)
  • primorial (~600)
  • "primitive roots" (~550)
  • Erdos (~550)
  • Hardy (~500)
  • Riemann (~500)
  • Lehmer (~500)
  • chess (~400)
  • eigen (~400)
  • Honaker (~400)
  • error-corrected (~400)
  • "upper bound" (~400)
  • truncatable (~400)
  • Goldbach (~360)
  • Poincare (314)
  • "polygonal numbers" (~275)
  • lucky (~260)
  • "number of iterations" (~250)
  • Thue-Morse (~250)
  • cellular automaton rules (~ 230)
  • greedy (~220)
  • "highly composite number" (~175)
  • Galois (~ 167)
  • square-free (~115)


I made a comment stemming from an OEIS hit for some bitwise work I was doing. Completely unexpected.

Yet another surprise hit

working with some A/D frequency response stuff led me to A140777 and A165355 (I was aware of neither). I likely will add comments to those.--Bill McEachen 23:40, 8 January 2015 (UTC)

  • also, in the course of my dallying, I looked at the specific approximation for ln(z) associated to the bilinear transformation (2 * (z − 1) / (z + 1). The crude text one is fine for z=0.7 to 1.0 but lousy for 0+ to 0.7. A cubic fit I did up (Excel) is superior and is 6.3853z313.476z2 + 10.481 * z − 3.3064. I use the std "z" associated with the discrete transfer function eqns. I know have my old A/D work online and available. I recently have come up with potential improvements for phase match.

Seq awaiting a next large term

Seq Gap


I submitted this as a variant it turns out of A000101...relates to prime gaps--Bill McEachen 15:42, 19 January 2015 (UTC)


A058043: Wolfram Alpha input: nextprime(n^2)-prime(PrimePi(n^2)), n=2 to 81

contributor spans (some may start earlier)

just something I was curious about ...

  • "Vincenzo Librandi" 2008-present
  • "Clark Kimberling" 2004-present
  • "Reinhard Zumkeller" 2002-present
  • "T. D. Noe" 2002-2014
  • "Robert G. Wilson"  ???-present
  • Greathouse 2005-preent
  • "Alois P. Heinz" 2008-present
  • "Jean-Fran├žois Alcover" 2011-present
  • "Amarnath Murthy" 2002-2010
  • "Colin Barker" 2007-present
  • "Benoit Cloitre" 2002-2014
  • "Vladeta Jovovic" 2000-2009
  • "David W. Wilson" 2000-2014
  • "Jonathan Vos Post" 2004-2014
  • "M. F. Hasler" 2006-2014
  • "Wolfdieter Lang" ???-present
  • "Michel Marcus" 2012-present
  • "Joerg Arndt" 2001-present
  • "Ralf Stephan" 2003-present
  • "Juri-Stepan Gerasimov" 2008-present
  • "Peter Luschny" 2005-present
  • "Michel Lagneau" 2010-present
  • "Vladimir Shevelev" 2007-present
  • "Lekraj Beedassy" 2000-2012
  • "Alonso del Arte" 2004-present
  • "Zhi-Wei Sun" 2008-present
  • "Don Reble" 2001-2014
  • "Jonathan Sondow" 2004-2014
  • "Gary Detlefs" 2000-2014
  • "Jud McCranie"  ???-2013

For comparison my span is 2003-present

A256620 (proposed)

relates to maximal runs of 6n+/-1 where primes are absent.--Bill McEachen 13:10, 5 April 2015 (UTC)

Rejected sequences and comments

My true observation on the Thue-Morse sequence was rejected. The comment was: The sequence is characterized by an expected mean of 0.5, though pair sums ratio to 1/2/2 for 0,1,2 respectively (unlike a coin flip, which yields 1/2/1).

A sequence (w/ technique) to spawn nicely volatile prime sequences from other sparse sequences was rejected as being an artificial construct. The sequence spawned from A002315 is: 17,41,2,3, 91393, 811, 947, 3, 2, 127, 5, 8071607521, 9369319, 5, 46083933182810391855077841108121860076301803820136729604319921407582209931247732827597272276147556142385931557060724704331319480811439873947611787983922003 , 7,2, 47, 5, 919, 3, 489133, 2, 8287, 2 , 43, 7 ,2 ,7 , 9285087769350986448116616132878186749607968459, 19, 5 ,7 ,5 ,61, ...

My comment on A151723 connecting it to 2 sequences was rejected (A197073, A022338). The 2 sequences characterize the three main "core" ON lines--Bill McEachen 13:10, 5 April 2015 (UTC)


I just saw Vladimir Shevelev's (VS)conjecture, which looked interesting. A249943 & A251621 have no Pari code. However, we do have code for A001223. So, restating VS: prime(n) = 19 + summation (A251621(i+4) ) i=9...n, n=8,9,...

or via VS's formula in A251621, prime(n) = 19 + summation (A001223(i-1) ) i=9...n, n=8,9,...

from the sequences, we see A001223(n) via forprime(p=1,50, print1(nextprime(p+1)-p, ", "))

The trick is to ensure we use no data "ahead" of prime(n). We know A001223(11)=6 from 37-31. This requires prime(12)=37. From VA, prime(12)=19+ A001223(8) + A001223(9) + A001223(10) + A001223(11)=19+4+6+2+6=37.

SO, it appears to me to require data "ahead" of what we seek, and so is not useful...

Let's try working from A098550 w/ RK's code (no explicit apriori primes required):

a(n, show=1, a=3, o=2, u=[])={n<3&&return(n); a+1==i&&print1("1, 2"); for(i=4, n, show&&print1(", "a); u=setunion(u, Set(a)); while(#u>1 && u[2]==u[1]+1, u=vecextract(u, "^1")); for(k=u[1]+1, 9e9, gcd(k, o)>1||next; setsearch(u, k)&&next; gcd(k, a)==1||next; o=a; a=k; break)); a}

A249943(n)=k makes use of A098550 thru k, where A098550 completes run to n. A251621 uses run lengths of A249943.

It's a bit complicated to follow, so we'll use an example. Akin to the example in A249943(7)=15,as A098550(15)=7 completes the coverage, so k=15, requiring thru A098550(15). However, to capture the run length requires thru A249943(9), which supplies A251621(7). But A249943(9) of course requires add'l A098550 entries, beginning a vicious cycle projecting forward...

Similarly, A249943(16), A098550(23)=13 completes the coverage, so k=13, requiring thru A098550(13). However, to capture the run length requires thru A249943(36), which supplies A251621(16).

So, there is some variable ptr> base n required from A098550, requiring data "ahead" of what we seek, and so again I conclude it is not useful...


more to come ....--Bill McEachen 14:58, 19 April 2015 (UTC)


after 100K, we need to see 19051 at some point (permutation...) the timing on the Pari script ~ terms^2.3 (rough) generally, the largest difference betw term and prime seen is 58207 (@ n=95016 which is encouraging). 19051 will require diff>~81K

Integer Notation

This just covers my approach for some of my work. I see Mr MF Hasler covers some of the same ground in his article here:M. F. Hasler/Representing large digits.

For non pure integer notation like decimal (or any radix I suppose), we have xxx.yyy where the power=0 begins to left of the decimal point and increases moving left. Similarly, the powers decrease to right of the decimal point from the inverse of power=1,2,....

I use a form www.zzz for integers of any base, with the imposed restriction of using digits 0-9 only. Power=1 begins to left of the decimal point, and any remainder <base to the right of same. One uses additive combinations as required (examples follow).

From hex, say #00e3 =227 base 10. As defined, this falls in the "gap" where a single notation is not possible for base 16, specifically values 160 through 256, since the notation goes from 9.15 to 10.0 (159 to 256). In my notation, 227 would be (9.15+4.4) (expanded 9*16^1+15 + 4*16^1+4 = 159+68=227 ).

Of course one clarification is that the base must be known/clear, as (9.15+4.4) is valid in any other radix>15 as well. Another nuance is there are multiple representations available, another could be 256-29= (10.0-1.13). This can be a plus or minus depending on ones requirements.

An advantage is that one can effectively limit the notational representation by merely increasing the base ( I gloss over some crucial intricacies here). A trival example might be moving from 9999999999999999.0 in some base, to merely just 1.0 in a much higher base.

I have not, and likely could not, mathematically prove every value can be represented for every radix but I think this is the case intuitively. When using VERY large bases one can see the desire for the restriction to digits 0-9.

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