# The sequence of digits in the decimal representation of powers of two?

Actually, a question: any sequence of decimal digits you can specify a degree of two in the decimal expansion where it will meet? And exactly how this exponent can be calculated? I'm interested in all sorts of materials relevant to the question.

01234567 found in...29485 = 185998256324337487767985099955302490665374438073335840272084270397484327790005864934072354482021720911607439665630887454808841855527882667163771398720576027402669580169039741850993940761617266622811864429881040671809439489622393109013165551200850833380933507325558571505564624577391929733959618246144977241913309659022912797548399276211570534729020491071351191290889636126433307695189713099137483570601661708273284507717023825388442891120735386162916399105393565702147204919204927037121724215287196788904608056905947536130449597358949449616051831907986767175594619583766525463617329127144503518763659942277269038905459230927989661987285329625187574824769288742763270245087547284539662518984216055365623621696956211603615545283516573789156694657671008997040572989022389295699721799750778506603701184419481263673489664211033837715782585254973569027982341874579420953999266744249299673279075763479994927498175002546738750955162901979077216015752598528464831465168854992680865829997423875909270885910965729572350830237045369449776277174455720770568210691006297077476042483939326526851979307090750800061956550603556501270224750819094942213629118099909177325337246786615531361534246491968360262811812215298071568437104822874348012370836228121589822780044987730811293226136550271694963934186396759877143359774015238471586730990536556332744717405897925895675069765677185890136024054302280347515023301882013138735012345674536443843500724073312200844070436544768321565730644790934123595660003672194718494935796078574313569457541209323393484644272286905883957862224606143980132378110678144627933700780795461464687395574955151452186658576874935551479315961297064331122140793650748872970125010994615841776971001836551719980645754836193958872457983938052160551666214065684852303128150087444204890759057273173850029313963600115272088506980655473444588000112953467252441483580477594551140995244830508723363302819687941270117760525826220041826619135638584326558176247505992357026743425616103447388318167511850110735269149887857412828805900122675014371794107967305858958751351419788686382412392518412100273742588751822095524644476110440217857585827426547924865922486611141726436838559414195948110593139374785498409121941376751557771017832079619769999600051786051653222771712610512883889823479670319462550051224474732291051859034097656799845251470736812586543469328564796307157930906773579236658091879601251498511749997235491825330900010269048713324079960509461271746921684816569470723524805341479469183318587968969248084022164275378401952871363267680996550172515684287413252616824865750445523554101027217128922653049387983014376322390036509760328751564360406787681572404381406365944853711323115460993534346396975045219006137336014139706260295262175593003673203123366570882309801070554543869201683365878375131751294372042353132928077347566058488076052266521358672510301765632
October 3rd 19 at 04:03
October 3rd 19 at 04:05
Hint:
n = loga an
a = b log ba
Next is to use the operations with series.

Start evidence for short sequences. Then with the help of deduction to find patterns in the number of possible degrees, etc.
October 3rd 19 at 04:07
Narrow down the task: we seek the power of two starts with the correct sequence numbers (if the sequence starts from zero, add to the beginning of the unit). Let's call it d for. Then we need to find x and y such that
d < 2x/10y < d + 1.

Let a = log210. Then
d < 2x — ay < d + 1.

Let's take a logarithm base 2:
log2d < x — ay < log2(d + 1)

It remains to choose x and y. Due to irrationality of a it is obvious that we will be able to find them.

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