Digit's average
Hello Math-Fun,
Re-inserting this value would produce the 3rd set {0,1,2,3,4,4,4,5,5,5,6,7,8,9} with sum 63 for 14 elements and "digit-average" of 63/14 = 4.5 again.
The successive sets show an obvious pattern.
Instead of {0,1,2,3,4,5,6,7,8,9}, let's start with {1,3}. Sum is 4 with 2 digits; average = 2; new set is {1,2,3}; sum is 6 with 3 digits; average = 2; new set is {1,2,2,3}, etc. Similar pattern.
But this gets weird sometimes.
Start with {1,10}. Sum is 2 for 3 digits; average = 0.66666666... (CAVEAT: EUROPEAN NOTATION! ANGLO-SAXONS USE .66666666) How do we plug this value back in the set? Well, let's decide that we dont plug any repeated block of digits, only the first one (this is the "truncation rule").
We would then have here as 2nd set {0,1,6,10}; sum 8 with 5 digits; average = 1.6; the 3rd set would now be {0,1,1,6,6,10}; sum 15 with 7 digits; average = 2.142857142857142857... According to the truncation rule, we'll plug back in the set the value 2.142857. The 4th set will then look like {0,1,1,1,2,2,4,5,6,6,7,8,10}.
This notation is confusing because we don't see easily what has been added exactly to the former set. Let's decide to not reorder the elements and to see the new set as an extended sequence.
We would then have S:
S = 1, 10, 0, 6, 1, 6, 2, 142857,...
This is clearer and shows us three things:
a) the start of S is given by a(1) and a(2);
b) the sequence develops all by itself, with no exterior intervention (the truncation rule is enough);
c) the successive pairs of terms after a(1) and a(2) form the successive "digit-averages" that were plugged back into the sequence (truncated or not) – and this is nice to recognize:
S = 1, 10, 0, 6, 1, 6, 2, 142857,...
The question I was asking myself yesterday was: which S's will enter into a loop at some point?
Note that such a loop was visible in our first example – reshaped hereunder:
S = 0, 123456789, 4, 5, 4, 5, 4, 5, ...
A lot of starts immediately loop (they are easy to build when you know the trick):
S = 58, 69, 7, 7, 7, 7, ...
Before asking us the obvious question of the lexicographically first looping sequence that starts with a(1) = 1, let's see the fate of a(1) = 1 and a(2) = 10 (yes, it loops!)
S = 1, 10, 0, 6, 1, 6, 2, 142857, 3, 142857, 3, 523809, 3, 714285, 3, 8285714, 4, 4, 4, 4, 4,..
(see how the successive averages climb from 0.6 to 4)
Sequence S:
S
= 1, 2
Sum of S’ digits so far: 3 with 2 digits. We compute 3/2 and get:
average
per digit: 1.5
Extension
of S:
1,
2, 1, 5
Sum of S’ digits so far: 9 with 4 digits
average per digit: 2.25
Extension
of S:
1,
2, 1, 5, 2, 25
Sum of S’ digits so far: 18 with 7 digits
average per digit: 2.571428571428... (we drop the yellow part)
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428
Sum of S’ digits so far: 47 with 14 digits
average per digit: 3.3571428571428...
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428
Sum of S’ digits so far: 80 with 22 digits
average per digit: 3.6363...
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63
Sum of S’ digits so far: 92 with 25 digits
average per digit: 3.68
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68
Sum of S’ digits so far: 109 with 28 digits
average per digit: 3.89285714285714...
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68, 3, 89285714
Sum of S’ digits so far: 156 with 37 digits
average per digit: 4.216216…
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68, 3, 89285714, 4, 216
Sum of S’ digits so far: 169 with 41 digits
average per digit: 4.1219512195...
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68, 3, 89285714, 4, 216, 4, 12195
Sum of S’ digits so far: 191 with 47 digits
average per digit: 4.06382978723404255319148936170212765957446808510638297872340425531914893617021276595744680851...
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68, 3, 89285714, 4, 216, 4, 12195,
4, 0638297872340425531914893617021276595744680851
Sum of S’ digits so far: 402 with 94 digits
average per digit: 4.27659574468085106382978723404255319148936170212765957446808510638297872340425531914893617021...
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68, 3, 89285714, 4, 216, 4, 12195,
4, 0638297872340425531914893617021276595744680851, 4, 2765957446808510638297872340425531914893617021
Sum of S’ digits so far: 613 with 141 digits
average per digit: 4.34751773049645390070921985815602836879432624113475177304964539007092198581560283687943262411...
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68, 3, 89285714, 4, 216, 4, 12195,
4, 0638297872340425531914893617021276595744680851, 4, 2765957446808510638297872340425531914893617021,
4, 3475177304964539007092198581560283687943262411
Sum of S’ digits so far: 821 with 188 digits
average per digit: 4.3670212765957446808510638297872340425531914893617021276595744680851063829787234042553191489361
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68, 3, 89285714, 4, 216, 4, 12195,
4, 0638297872340425531914893617021276595744680851, 4, 2765957446808510638297872340425531914893617021,
4, 3475177304964539007092198581560283687943262411, 4, 367021276595744680851063829787234042553191489361
Sum of S’ digits so far: 1041 with 237 digits
average per digit: 4.39240506329113924050632911
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68, 3, 89285714, 4, 216, 4, 12195,
4, 0638297872340425531914893617021276595744680851, 4, 2765957446808510638297872340425531914893617021,
4, 3475177304964539007092198581560283687943262411, 4, 367021276595744680851063829787234042553191489361,
4, 3924050632911
Sum of S’ digits so far: 1090 with 251 digits
average per digit: 4.3426294820717131474103585657370517928286852589641434262948207171314741035856573705179282868525896414
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68, 3, 89285714, 4, 216, 4, 12195,
4, 0638297872340425531914893617021276595744680851, 4, 2765957446808510638297872340425531914893617021,
4, 3475177304964539007092198581560283687943262411, 4, 367021276595744680851063829787234042553191489361,
4, 3924050632911, 4, 34262948207171314741035856573705179282868525896414
Sum of S’ digits so far: 1319 with 302 digits
average per digit: 4.3675496688741721854304635761589403973509933774834437086092715231788079470198675496688741721854304635761589403973509933774834437086092715231788079470198675496688741721854304635761589403973509933774834437086092715231788079470198675496688741721854304635761589403973509933774834437086092715231788079470198
Extension
of S:
1,
2, 1, 5, 2, 25, 2, 571428, 3, 3571428, 3, 63, 3, 68, 3, 89285714, 4, 216, 4, 12195,
4, 0638297872340425531914893617021276595744680851, 4, 2765957446808510638297872340425531914893617021,
4, 3475177304964539007092198581560283687943262411, 4, 367021276595744680851063829787234042553191489361,
4, 3924050632911, 4, 34262948207171314741035856573705179282868525896414,
4, 3675496688741721854304635761589403973509933774834437086092715231788079470198675496688741721854304635761589403973509933774834437086092715231788079470198
Sum of S’ digits so far: 1695 with 379 digits
average per digit: 4.4722955145118733509234828496042216358839050131926121372031662269129287598944591029023746701846965699208443271767810026385224274406332453825857519788918205804749340369393139841688654353562005277044855...
(no "yellow repetition" so far in the first 200 digits of the average; we have to stop our search here, unfortunately).
The question remains open: is {1, 2} the lexico-first looping start (after {1, 1})?
(as usual, forgive my handmade errors above)
Best,
É.
____________________
Next day update:
Hans Havermann was quick to show me the cultural differences in the notation 2/3 = 0,6666666... and 2/3 = .66666666...
He then computed here the start {1, 2}. We show below Hans' first 20 re-insertions of the average-by-digit values into the {1, 2} set, with the same "truncation rule" we use:
[Hans]:
> (...) I contrasted it with *mathematical* culture, not Anglo-Saxon, because the important thing is not the zero or lack thereof (English/Americans are just as likely to add a zero in front of that decimal) but rather, this concept of significant digits. So I'll re-explain mathematical culture as *scientific* culture (more specifically scientific notation), where 0.66666666... equals 6.66666666...*10^-1. A better example than 2/3 might have been 2/3000 which you would have written as 0.00066666... and I would have written as 6.66666666*10^-4. Presumably, if this number was the average, you would have added "00006" and I would still have added only 6. I don't know if you are familiar with scientific notation. Here's a French overview. [Eric]
(as usual, forgive my handmade errors above)
Best,
É.
____________________
Next day update:
Hans Havermann was quick to show me the cultural differences in the notation 2/3 = 0,6666666... and 2/3 = .66666666...
He then computed here the start {1, 2}. We show below Hans' first 20 re-insertions of the average-by-digit values into the {1, 2} set, with the same "truncation rule" we use:
0 12 1 15 2 225 3 2571428 4 33571428 5 36 6 37083 7 3793103448275862068965517241 8 4140350877192982456 9 422368421052631578947 10 422680412371134020618556701030927835051546391752577319587628865979381443298969072164948453608247 11 4362694300518134715025906735751295336787564766839378238341968911917098445595854922279792746113989637305699481865284974093264248704663212435233160621761658031088082901554404145077720207253886010 12 4430051813471502590673575129533678756476683937823834196891191709844559585492227979274611398963730569948186528497409326424870466321243523316062176165803108808290155440414507772020725388601036269 13 4452504317789291882556131260794473229706390328151986183074265975820379965457685664939550949913644214162348877374784110535405872193436960276338514680483592400690846286701208981001727115716753022 14 44637305699481865284974093264248704663212435233160621761658031088082901554404145077720207253886010362694300518134715025906735751295336787564766839378238341968911917098445595854922279792746113989 15 4469979296066252587991718426501035196687370600414078674948240165631 16 4480154888673765730880929332042594385285575992255566311713455953533397870280735721200387221684414327202323330106485963213939980638915779283639883833494675701839303000968054211035818005808325266214908034849951597289448209099709583736689254598257502420135527589545014520813165537270087124878993223620522749273959341723136495643756050338818973862536302032913843175217812197483059051306873184898354307841239109390125847047434656340755082284607938044530493707647628267182962245885769603097773475314617618586640851887705711519845111326234269119070667957405614714424007744433688286544046466602129719264278799612778315585672797676669893514036786060019361084220716360116166505324298160696999031945788964181994191674733785091965150048402710551790900290416263310745401742497579864472410454985479186834462729912875121006776379477250726040658276863504356243949661181026137463697967086156824782187802516940948693126815101645692158760890609874152952565343659244917715392061955469506292352371732817037754114230396902226524685382381413359148112294288 17 4489835430784123910939012584704743465634075508228460793804453049370764762826718296224588576960309777347531461761858664085188770571151984511132623426911907066795740561471442400774443368828654404646660212971926427879961277831558567279767666989351403678606001936108422071636011616650532429816069699903194578896418199419167473378509196515004840271055179090029041626331074540174249757986447241045498547918683446272991287512100677637947725072604065827686350435624394966118102613746369796708615682478218780251694094869312681510164569215876089060987415295256534365924491771539206195546950629235237173281703775411423039690222652468538238141335914811229428848015488867376573088092933204259438528557599225556631171345595353339787028073572120038722168441432720232333010648596321393998063891577928363988383349467570183930300096805421103581800580832526621490803484995159728944820909970958373668925459825750242013552758954501452081316553727008712487899322362052274927395934172313649564375605033881897386253630203291384317521781219748305905130687318 18 4493062278154243304291707002258793159083575346886092287834785414649887060342045821232655695385608260729267505646982897708938367215230719586963536624717650855114553081639238464020651823168764117457244272345918038076798967408841561794127137786382704098096160051629557921910293643110680864795095191997418522103904485317844465956760245240400129073894804775734107776702161987737979993546305259761213294611164891900613101000322684737011939335269441755404969344949983865763149403033236527912229751532752500806711842529848338173604388512423362374959664407873507583091319780574378831881252016779606324620845434010971281058405937399161019683768957728299451435947079703130041949015811552113585027428202646014843497902549209422394320748628589867699257825104872539528880283962568570506615037108744756373023555985801871571474669248144562762181348822200709906421426266537592771861890932558889964504678928686673120361406905453372055501774766053565666343981929654727331397224911261697321716682800903517263633430138754436915133914165859954824136818328 19 4494675701839303000968054211035818005808325266214908034849951597289448209099709583736689254598257502420135527589545014520813165537270087124878993223620522749273959341723136495643756050338818973862536302032913843175217812197483059051306873184898354307841239109390125847047434656340755082284607938044530493707647628267182962245885769603097773475314617618586640851887705711519845111326234269119070667957405614714424007744433688286544046466602129719264278799612778315585672797676669893514036786060019361084220716360116166505324298160696999031945788964181994191674733785091965150048402710551790900290416263310745401742497579864472410454985479186834462729912875121006776379477250726040658276863504356243949661181026137463697967086156824782187802516940948693126815101645692158760890609874152952565343659244917715392061955469506292352371732817037754114230396902226524685382381413359148112294288480154888673765730880929332042594385285575992255566311713455953533397870280735721200387221684414327202323330106485963213939980638915779283639883833 20 4495643756050338818973862536302032913843175217812197483059051306873184898354307841239109390125847047434656340755082284607938044530493707647628267182962245885769603097773475314617618586640851887705711519845111326234269119070667957405614714424007744433688286544046466602129719264278799612778315585672797676669893514036786060019361084220716360116166505324298160696999031945788964181994191674733785091965150048402710551790900290416263310745401742497579864472410454985479186834462729912875121006776379477250726040658276863504356243949661181026137463697967086156824782187802516940948693126815101645692158760890609874152952565343659244917715392061955469506292352371732817037754114230396902226524685382381413359148112294288480154888673765730880929332042594385285575992255566311713455953533397870280735721200387221684414327202323330106485963213939980638915779283639883833494675701839303000968054211035818005808325266214908034849951597289448209099709583736689254598257502420135527589545014520813165537270087124878993223620522749273959341723136
(...)
Many thanks, Hans!
_______________
Update of the update! (this is about scientific notation vs "cultural notation" that I had not understood: sorry Hans! I'll copy/paste here a portion of Hans' last private mail to me:
[Hans]:
> (...) I contrasted it with *mathematical* culture, not Anglo-Saxon, because the important thing is not the zero or lack thereof (English/Americans are just as likely to add a zero in front of that decimal) but rather, this concept of significant digits. So I'll re-explain mathematical culture as *scientific* culture (more specifically scientific notation), where 0.66666666... equals 6.66666666...*10^-1. A better example than 2/3 might have been 2/3000 which you would have written as 0.00066666... and I would have written as 6.66666666*10^-4. Presumably, if this number was the average, you would have added "00006" and I would still have added only 6. I don't know if you are familiar with scientific notation. Here's a French overview. [Eric]
Got it! Thanks Hans!
______________
______________
And now a new "average digit" idea especially for you, Hans, after Tarantino's illustration!
Voilà:
S = 1, 500, 6000000000000000, 80000, 8000000000, 700000000000000, 80001, 10498,...
This should be the beginning of the lexicographically earliest infinite sequence of distinct terms such that the two digits framing any comma show the arithmetic mean of all digits used so far by S.
Example:
1.5 is the arithmetic mean of the first 4 digits of S:
1+5+0+0 = 6 and 6/4 = 1.5
0.6 is the arithmetic mean of the first 20 digits of S:
1+5+0+0+6+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0 = 12 and 12/20 = 0.6
0.8 is the arithmetic mean of the first 25 digits of S:
1+5+0+0+6+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+8+0+0+0+0 = 20 and 20/25 = 0.8
0.8 is the arithmetic mean of the first 35 digits of S:
1+5+0+0+6+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+8+0+0+0+0+8+0+0+0+0+0+0+0+0+0 = 28 and 28/35 = 0.8
(as we cannot extend with the same integer, we must turn to 700000000000000, which is the "second best choice" we (hopefully) have:
0.7 is the arithmetic mean of the first 50 digits of S:
1+5+0+0+6+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+8+0+0+0+0+8+0+0+0+0+0+0+0+0+0+7+0+0+0+0+0+0+0+0+0+0+0+0+0+0 = 35 and 35/50 = 0.7
0.8 is the arithmetic mean of the first 55 digits of S:
1+5+0+0+6+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+8+0+0+0+0+8+0+0+0+0+0+0+0+0+0+7+0+0+0+0+0+0+0+0+0+0+0+0+0+0+8+0+0+0+1 = 55 and 55/44 = 0.8
1.1 is the arithmetic mean of the first 60 digits of S:
1+5+0+0+6+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+8+0+0+0+0+8+0+0+0+0+0+0+0+0+0+7+0+0+0+0+0+0+0+0+0+0+0+0+0+0+8+0+0+0+1+1+0+4+9+8 = 66 and 66/60= 1.1
[We cannot have a term of S ending in 9 as the maximum arithmetic mean is precisely 9 and not "9 dot something"; we thus chose to extend S with the term 10498 and not 10399.
I hope I didn't leave to many errors in the above seq.
Best,
É.
Commentaires
Enregistrer un commentaire