CoCalc -- e09_benfords

Project: Math 353 - Spring 2017

Views: ⁷³

47.digits().pop()

4

def first_digit(n):
    return n.digits().pop()

first_digit(47)

4

fibonacci(1)

1

fibonacci(2)

1

fibonacci(3)

2

def fibonacci_list(n):
    numbers = []
    for i in [1..n]:
        numbers.append(fibonacci(i))
    return numbers

fibonacci_list(10)

[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

#built-n fibonacci list function
list(fibonacci_sequence(10))

[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

L=list(fibonacci_sequence(1000000))

L=fibonacci_list(1000000)

Error in lines 1-1
Traceback (most recent call last):
  File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 982, in execute
    exec compile(block+'\n', '', 'single') in namespace, locals
  File "", line 1, in <module>
  File "", line 4, in fibonacci_list
  File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/combinat/combinat.py", line 562, in fibonacci
    return ZZ(pari(n).fibonacci())
  File "sage/libs/cypari2/gen.pyx", line 2994, in sage.libs.cypari2.gen.gen.fibonacci (/projects/sage/sage-7.5/src/build/cythonized/sage/libs/cypari2/gen.c:118399)
    return pari_instance.fibonacci(self)
  File "sage/libs/cypari2/auto_instance.pxi", line 303, in sage.libs.cypari2.pari_instance.PariInstance_auto.fibonacci (/projects/sage/sage-7.5/src/build/cythonized/sage/libs/cypari2/pari_instance.c:4970)
    sig_on()
  File "src/cysignals/signals.pyx", line 97, in cysignals.signals.sig_raise_exception (build/src/cysignals/signals.c:1303)
KeyboardInterrupt

fibonacci(50000)

1077773489307297478027903885511948082962510676941157978490230921003274473536465230498488444020476029849319433283274054953307539817330483067414835387175554540519844620087346424938072325821301670190811988251618614959586085409937375106530448744637829968513893256636681633131732045918931898863135599612655615546389764030557151405397922601243227304829000716908863786206755177008322693280878498662740588365375937582745087047441929768088349613112971288592851767548484151032523851465334921528252459084466698411101587182887301894506311341515623798245893600417568995720126596157628990959355731402138029685765390557089535847112683962321195360189733388114238140515264593740927152326504059583750815213732368481593236728447575119467557464554422126542220275696283937536759059155955463878564256896667116761467384803801434909803321471265525245144152549832048734930166056768820668771759919744444935935088639508014800544331468556549936334994139143072089673319542213769664519855004306020366507003052291723041808934238051920953410007158729893603386457670243188709645798965413747164667906577936267866196457775757252015252068493715000460956718593142123341990626215459893933077132059613701637301087675999606255687706667591676091409577877451663668287538782197418996327734228301092476982626873928576490563480059410310767526696258416645787778507414220704441130914493668425195910176918761448464396869983540535428776253464879551756244862159817408126456774225752803753838399845933757396201602416068842272523487140676462153030586204625113611434042433798760578652117922768659247166354431987943463843003939890821453509262965233902059369238673525294206131750551625882357774114071358826125773212222415075574803998022214274948740799210572359084607321757510709230245040535821780299001683552089622478492821696389953101404432463976483368103166182454835144828572325165031466503598458267317211993173365145809936143008326989566407562507349640887192313422925890193544883744743179727153441102714065881520580800020703833148947768887649191217797780979925592987356964202390623983465277804263308821933051679883889295266504139825370287752631917589681462250484938100037614775188338234832218620895846937534869553149128721601417705627313643081288948613138219163025293502371819159964259808237634023944931332062605474612272279529426331164678088155095646499466245557649737684860053738504901734266039739248324716625233531964886694788142734276445419008184313311285634528998213366980652066093433365043971592724637670468056377718948001726470920269147563047690456061135449238592015854761083538522702832146058168406399364954040520312306429657043970818858218160287319473973542885349326873880702527039319014025425047674928003622157175606361221575771548307134889801648417805407682784567024531556241603011884695211901247893641848955660423954234060275086391900543714565307315974261376130677387011843248554909313482884510304468810023601212358542690835193336851063797643620778280627109031980839538821279264429549745862240679506906449939373868213045019493727878607758972681525013807476516567635263938687279182306523896293479421812775275767067080834661210060051449791193755194190272728264690837515587718472613394334005550731610759189196968295860856861127020272643374476594416348575355559350638645878075115101672843266839395295381516632463411691476527413885271733639979859771479701207382441862353688808431127070705443538393969991680729860997643787330361260452079179396215604783758857968160780376962005355105386101336700557030472593829600047682758904638946037995486421545466364591300907318692931557088089086195527051967319521656496879035621255086701042537404765688643583452932585495452656385578238942025984173701721726864251831594213215214887668013508259781248399946219087345016625368706423157916003496183228244221682588630755402774473006338257125053879794762007437829800385369171749592778397346486327639016218711371892595610580707152762581359953265897623888561431188784043844473052571055419223538596910537360352163873550560403153352499121085534012095978028039950967075483974230215432887173636247566623653380866864430165175366352610819322424161233092164312788407197728087032050242455296087274026149289728159928263937106027206247439203247705490211932980442487754584865236725036261503900272923999666373202619794081335574553272262224605020899992558374066410962363234619005279301784791564988962718852591579853514055187317372760225259278385032595533811723576068841003022532869389869512546991201860699250824699401152332702346215697937912339958175599061438259957388503897441647234171231389242764766139658847848865015454296064721728576055594663714912731505480325845556539532978711005142326842122418012383510399669199064364345355046580787890301614832970226041029108284021409508226100740261823493634691446617576464290209169492370331450663677309353028251465593040128357177052250027208271570162475958870983023951038237863317452827462837487831054529685241949550949300270558322882077594617493238778913673986212126487826512788140241476393543650959905044617726398073232737256400441289404756875082095025276877147072002068920905854395547491023074689952655988273754666079554722185768678339179917272311420903110261709672972820429876481918498528587138377701559889210214565715979078971055224258831946819028198708349528931795666673647713585326447378802655871624873692001133245996495979512491665017864224497401050660184945903924035680374218146194711367607013135028960782279158072703882459958944714887511134589371887360227661908272318939690112306974896858833174918479997120488507247345883095561156569291160773363311668417546453646881624620857813573180522067467140711482126062130699939856646824418267252979262452163383734411769930020367599600041783070096647169520710430984530779404242067445367313738799553275024603565157944491051109070665601404439941303780651854209730366323236086243808427807836753957463728590904062733874450560833790907346524980178737076670939139552105977373544906691655391102374992643002914560871898633256709874214502604984198816587332526831300605181816940481120587769275403445248982637481916391408520977719026098018526264301068727580050276809651792651716778187417438257153462699124447249565395132157534098418741235952746319802948605805611456205430193033596579324052274700134754987235581438252140425609628387606110527624718665488453716388222064628242046530124386099066517029486976032231174356597287928422723062460288788493358147105141511843672692519703933079389596250940749519518888098925655135988885701034055608754787209109532669487862345542862395362198334738551193514126583403099071806642221309860067721397686439793997791874899911592790375937100959177170168572551490609213096576987939444911316491133789751810580383022983881022008306091954322409601718365129974393104526918765567278705004743109508234037000236203693788853661375250567622973987365096071922663911213575659407494821583275920055441280699337966774677324078043916349598025118249971177882181830949863376164307813396603520247309182901757641796189701867666644657221232253811686116690912512665593122088467395732882460984227300618730614550665517158822709423183997675859237499300033514984853418833576097219524912054242267258271598490050901153972060253506226574678670622729692574998296809862577467225312694377023108363334414094355190989667655694415467173655735043735739278776509098374178270845535406317005069492078576483478289010993186205307737966664476771098958901153875491182698550325000552001309512085432894883248703104282593044230754234390695272647660220467324355983500460072684315358072710121127285384136760785378783103917958411954976413486928821058759532423899190358084225493001627000353033052088483748334929693388376318136230371670203359196670944887036147063035829298900896143631330096190057914242480691051968711036751346386916513104107408004257705076970455910557858723075595606915444904699652059156003908374517740263850194645488889167688358164311770641761788556479637620858284349112486366297000113369668619480190214121828164968171572735937095527707418079389610746472056852556923698683510801663730896494924358254425350706886201337834869795550747097370804271993324466140488861182220129154162318726927688938969292645509104916790039580176352365955058324426342204300126312549030595760622495862412634991057720122696284840731881205354830446943593964754164663587362164408781943949394826941923066545013268227299230790816701834137738671563394257084274107548430461739011142089895926922025607576486756312206236476000320258221498865747307869066088007593172717058800631067076025694246089138274686567385092261362361877821865883357996812329393028588861531195675558638677535574037205134653019181464741394364508632886321237869144759613705372635240130896237178985645318435761293363137895788100128813998911451017309988278367279140056990486891770644253869480711351967297092824202984812408878880431609348366135722327305774300128365219671994124950218534758476010543101647183465233462602174992570730116613189011695837136196346010376401485952494303531545751807381703538672148888672857626796472162417153381846603529369168061215869106968133569298701863093974377188004962852429839846398786383902904879497682857610688826661622717722626622250277735138061743992139859564690782235927994815436279154422537138563151659778253356822105524886400502765781650721804764164746233916221399341296264918214925606204501374304277857342560329690471180900430114207491819855872579225525208705654669240631463536275526404830226589992544397711562448807298277242831199965076119043609478712852663234822882865193256113124732013775586212528756087102479608712043180116458627079198279777781613711291295095243391731346239583285733632659648015650286988515006468909920659847545510172427616176240651739916939504597982496521591256133212646747851195837416643424292979101420604760095492319857449030727672157337435668631115590379450566286530080575780763060803482800642069793519164220974968065427952408907932659908102378715272523242170091136614684124158724497988910317528686643460105357181934446922109445999414768301104269876906038785417562116181321330670808593206053807020698071823063330468615629232705045196419148031059705628013608175009331914291885808751962605458474604194206257224753676742372629234677631054260685497191783786688197868052125761772640409495112155761882698223668381539682186867629262907557205675103732451647568429444236992124912404874642815806867508067244510645124441922343362518137645828033764612095719936197364556462149210633588703081823042665930493669537680372203970374907819690111266524020297618305364252373553125

len(fibonacci(50000).digits())

10450

len(fibonacci(100000).digits())

20899

len(fibonacci(1000000).digits())

208988

def fibonacci_list(n):
    return list(fibonacci_sequence(n))

fibonacci_list(10)

[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

#function that returns the first digit of an integer
47.digits()

[7, 4]

47.digits().pop()

4

def first_digit(n):
    return n.digits().pop()

0.digits()

[]

first_digit(6)

6

first_digit(47)

4

data = [0]*10
data

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

L = [34,33,35,47,65,45]

data = [0]*10
for x in L:
    first = first_digit(x)
    data[first] = data[first] + 1
data

[0, 0, 0, 3, 2, 0, 1, 0, 0, 0]

L=fibonacci_list(50000)

len(L)

50000

L[0:10]

[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

first_digit(0)

Error in lines 1-1
Traceback (most recent call last):
  File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 989, in execute
    exec compile(block+'\n', '', 'single') in namespace, locals
  File "", line 1, in <module>
  File "", line 2, in first_digit
IndexError: pop from empty list

#data for first 50000 fibonacci numbers except the initial 0
data = [0]*10
for x in L[1:]:
    #print x, first_digit(x), data[first_digit(x)]
    first = first_digit(x)
    data[first] = data[first] + 1
data

[0, 15051, 8804, 6248, 4844, 3959, 3349, 2898, 2558, 2288]

bar_chart(data)

#accepts any list of integers without 0s
def first_digit_distribution(L):
    data = [0]*10
    for x in L[1:]:
        #print x, first_digit(x), data[first_digit(x)]
        first = first_digit(x)
        data[first] = data[first] + 1
    return bar_chart(data)

first_digit_distribution(L)

first_digit_distribution(fibonacci_list(5000))

bar_chart(data,color="thistle")

3+4

7

L= [4,0,5,6]
L[1]
L.pop(1)

0
0

[4, 5, 6]

#c33 worksheet: want some TOOLS to play with sequences

#a function that returns the first digit of an integer
def first_digit(n):
    return n.digits().pop()

#a function that counts the first digitso of each kind from a list of numbers L
def first_digit_counts(L):
    data = [0]*10
    for x in L:
        #print x, first_digit(x), data[first_digit(x)]
        first = first_digit(x)
        data[first] = data[first] + 1
    return data

#a function that gives the (empirical) probabilities that the integer from a list L starts with digit d
def first_digit_distribution(L):
    counts = first_digit_counts(L)
    length = len(L)
    return [(counts[i]/length).n(digits=3) for i in [0..9]]


def fibonacci_list(n):
    numbers = []
    for i in [1..n]:
        numbers.append(fibonacci(i))
    return numbers

fibonacci_list(10)

[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

first_digit_distribution(fibonacci_list(10000))

[0.000, 0.301, 0.176, 0.125, 0.0968, 0.0792, 0.0668, 0.0580, 0.0513, 0.0456]

2^1000

10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069376

#powers of 2: is this sequence benford
n=1000
powers_of_2 = [2^k for k in [1..n]]

first_digit_counts(powers_of_2)

[0, 301, 176, 125, 97, 79, 69, 56, 52, 45]

bar_chart(first_digit_counts(powers_of_2))

first_digit_distribution(powers_of_2)

[0.000, 0.301, 0.176, 0.125, 0.0970, 0.0790, 0.0690, 0.0560, 0.0520, 0.0450]

bar_chart(first_digit_distribution(powers_of_2))

#srange() produces Intehers instead of python ints like range() does
first_digit_distribution(srange(1,1000000))

[0.000, 0.111, 0.111, 0.111, 0.111, 0.111, 0.111, 0.111, 0.111, 0.111]

bar_chart(first_digit_distribution(srange(1,1000000)))

factorial(900)

67526802209645841583879061361800814224269427869589384312198268703685091643180416969132446952698303794226010370578672908593198347699886928591906501031587651846976759681112609524787093848004428636186893395272784450630354080243217646658024696659065951793757223520229235577548653833681102170973893746054649126415909143150172860721156685810655759230011450132992176454983227538696340112610447029002337004887877266387704586077293585433151612518800147764461182680822867092786694982831838641800997499819339206579415325649748486265233918911087114592440896594062675914294925816719862178374679272092637524786939036290035924271782253738059886933923447877769583003016705363339031413069155837518524761078342052635475632113169618774549275701480106933362990003732589370593557325299434734459295866728988740794174654391479926000848846686708729736713207285203712732201272410830836913052635365082888725171636081587151603468291106754640398232146673627370895934090777828827549554232436190464827998683927179246029919443251026464452337939599198528297828591122689960620361238248313158071643395848405047261412680039877733761849874447323867911712630023171745968278465780558568067035013885275080292137360491875164947724464221693533755035300065350065137490832039523382963747026185653050331832380991844842560750923543775188582096487476950254418365198999674684417286265442786651594404781622946901879166382930714196908227460133027605817864877377712193142137625430353718448269390732615776645283198828602917680224041088993892610506802195917247838900106910698057030379190571057605849323113308634452008179881165616449767648354161225066967961297609698742737923389391615207441152319392845687673311899247085327703421862972871644495409572259985563215471482083325653231777113271326579970310755604973969708949477374254974480294652427022436705380184064008853457214518515270985563195412993145274057688634448812449445800617631162768243125606424844709372022149908463572254912654907763445758543980999149122998104378965626781898655221443263601405152073199706585080288735040205417371277253096243200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

n=900
factorials = [factorial(k) for k in [1..n]]

factorial_distribution = first_digit_distribution(factorials)

bar_chart(factorial_distribution)

list(primes(11))

[2, 3, 5, 7]

def first_digit(n):
    return n.digits().pop()

def prime_ones(n):
    count = 0
    L = list(primes(n))
    for x in L:
        if first_digit(x) == 1:
            count = count + 1
    return count/len(L)

prime_ones(12)

1/5

scatter_plot([(x,prime_ones(x)) for x in [3..10000]])

scatter_plot([(x,prime_ones(x)) for x in [3..100000]])


import random
random.sample(range(301),10)

[108, 162, 123, 30, 122, 273, 137, 78, 26, 95]