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A088995
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Least k>0 such that the first n digits of 2^k and 5^k are identical.
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1
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5, 98, 1068, 1068, 127185, 3053508, 22349997, 73482154, 961700165, 961700165, 248045160416, 1404948108914, 13801435928724, 243632550916041, 5729166542536373, 5729166542536373, 331484151442699072
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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]
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LINKS
| T. Sillke, Powers of 2 and 5 Puzzle
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EXAMPLE
| a(2) = 98: 2^98 = 316912650057057350374175801344 and 5^98 = 315544362088404722164691426113114491869282574043609201908111572265625.
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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]
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CROSSREFS
| Cf. A088935.
Cf. A010467.
Sequence in context: A202516 A062538 A053980 * A093749 A197474 A147539
Adjacent sequences: A088992 A088993 A088994 * A088996 A088997 A088998
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KEYWORD
| base,nonn
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AUTHOR
| Lekraj Beedassy (blekraj(AT)yahoo.com), Dec 01 2003
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EXTENSIONS
| Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 02 2003
More terms - J. Mulder (jasper.mulder(AT)planet.nl), Feb 03 2010
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