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A088995 Least k>0 such that the first n digits of 2^k and 5^k are identical. 1
5, 98, 1068, 1068, 127185, 3053508, 22349997, 73482154, 961700165, 961700165, 248045160416, 1404948108914, 13801435928724, 243632550916041, 5729166542536373, 5729166542536373, 331484151442699072 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The number of matching first digits of 2^n and 5^n increases with n and forms the sequence 3,1,6,2,2,7,7,6,6,... which approaches Sqrt(10).

Numbers are half of the denominator of some convergent Log[2]/Log[10] [From J. Mulder (jasper.mulder(AT)planet.nl), Feb 03 2010]

LINKS

Table of n, a(n) for n=1..17.

T. Sillke, Powers of 2 and 5 Puzzle

EXAMPLE

a(2) = 98: 2^98 = 316912650057057350374175801344 and 5^98 = 315544362088404722164691426113114491869282574043609201908111572265625.

MATHEMATICA

L2 = N[ Log[ 10, 2 ], 50 ]; L5 = N[ Log[ 10, 5 ], 50 ]; k = 1; Do[ While[ Take[ RealDigits[ 10^FractionalPart[ L2*k ] ][[ 1 ] ], n ] != Take[ RealDigits[ 10^FractionalPart[ L5*k ] ][[ 1 ] ], n ], k++ ]; Print[ k ], {n, 1, 10} ]

L2 = N[ Log[ 10, 2 ], 50 ]; L5 = N[ Log[ 10, 5 ], 50 ]; k = 1; Do[ While[ Take[ RealDigits[ 10^FractionalPart[ L2*k ]][[ 1 ]], n ] != Take[ RealDigits[ 10^FractionalPart[ L5*k ]][[ 1 ]], n ], k++ ]; Print[ k ], {n, 1, 7} ]

f[n_, k_] := {Floor[ 10^(k - 1 + N[FractionalPart[n Log[5]/Log[10]], 20])], Floor[10^(k - 1 + N[FractionalPart[n Log[2]/Log[10]], 20])]} Flatten@Block[{$MaxExtraPrecision = \[Infinity]}, Block[{l = Denominator /@ Convergents[Log10[2], 1000]}, Array[k \[Function] l[[Flatten@Position[f[ #/2, k] & /@ l, {x_, x_}, {1}, 1]]]/2, 20]]] [From J. Mulder (jasper.mulder(AT)planet.nl), Feb 03 2010]

CROSSREFS

Cf. A088935.

Cf. A010467.

Sequence in context: A062538 A053980 A215299 * A093749 A197474 A301307

Adjacent sequences:  A088992 A088993 A088994 * A088996 A088997 A088998

KEYWORD

base,nonn

AUTHOR

Lekraj Beedassy, Dec 01 2003

EXTENSIONS

Edited by Robert G. Wilson v, Dec 02 2003

More terms - J. Mulder (jasper.mulder(AT)planet.nl), Feb 03 2010

STATUS

approved

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Last modified June 25 03:50 EDT 2019. Contains 324338 sequences. (Running on oeis4.)