A018802 -id:A018802 - cp's OEIS frontend

A018856 2^a(n) is the smallest power of 2 beginning with n.

0, 1, 5, 2, 9, 6, 46, 3, 53, 10, 50, 7, 17, 47, 77, 4, 34, 54, 84, 11, 31, 51, 61, 81, 8, 18, 38, 48, 68, 78, 98, 5, 25, 35, 45, 55, 75, 85, 95, 12, 22, 32, 42, 145, 52, 62, 72, 82, 92, 102, 9, 19, 29, 39, 142, 49, 59, 162, 69, 79, 89, 192, 99, 6, 16, 119, 26

Offset: 1

David W. Wilson

A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems With Elementary Solutions, Vol. 1, pp. 29, 199-200, Prob. 91a, Dover, NY, 1987.

Daniel Mondot, Table of n, a(n) for n = 1..32699

Cf. A018802.
Cf. A100129 (a(n) = n).
Cf. A030000, A000079.

import Data.List (isPrefixOf, findIndex)
import Data.Maybe (fromJust)
a018856 n =
   fromJust $ findIndex (show n `isPrefixOf`) $ map show a000079_list
-- Reinhard Zumkeller, Aug 04 2011

f[n_] := Block[{k = 1, m = Floor[ Log[10, n]]}, While[ Log[10, 2^k] < Floor[ Log[10, n]], k++ ]; While[ Quotient[2^k, 10^(Floor[k*Log[10, 2]] - m)] != n, k++ ]; k]; f[1] = 0;; Array[f, 73] (* Robert G. Wilson v, Jun 02 2009 *)

from itertools import count
def aupton(terms):
    adict, pow2 = dict(), 1
    for i in count(0):
        s = str(pow2)
        for j in range(len(s)):
            t = int(s[:j+1])
            if t > terms:
                break
            if t not in adict:
                adict[t] = i
        if len(adict) == terms:
            return [adict[i+1] for i in range(terms)]
        pow2 *= 2
print(aupton(67)) # Michael S. Branicky, Apr 08 2023

A018869 Smallest power of 9 that begins with n.

Original entry on oeis.org

1, 282429536481, 387420489, 4782969, 59049, 6561, 729, 81, 9, 109418989131512359209, 11972515182562019788602740026717047105681, 12157665459056928801, 1350851717672992089

Offset: 1

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 1..369

Cf. A018802 (k=2), A018857 (k=3), A018859 (k=4), A018861 (k=5), A018863 (k=6), A018865 (k=7), A018867 (k=8), this sequence (k=9).

a(n) = 9^A018870(n). - Seiichi Manyama, Jan 29 2017

a(9^n) = 9^n for n >= 0. - Seiichi Manyama, Jan 29 2017

A018861 Smallest power of 5 that begins with n.

Original entry on oeis.org

1, 25, 3125, 48828125, 5, 625, 78125, 88817841970012523233890533447265625, 9765625, 108420217248550443400745280086994171142578125, 11920928955078125, 125, 1387778780781445675529539585113525390625, 1490116119384765625, 15625

Offset: 1

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 1..326

Cf. A018802 (k=2), A018857 (k=3), A018859 (k=4), this sequence (k=5), A018863 (k=6), A018865 (k=7), A018867 (k=8), A018869 (k=9).

a(n) = 5^A018862(n). - Seiichi Manyama, Jan 29 2017

a(5^n) = 5^n for n >= 0. - Seiichi Manyama, Jan 29 2017

A171132 The smallest exponent k such that the digit n appears at two adjacent places in the decimal representation of 2^k.

Original entry on oeis.org

53, 40, 43, 25, 18, 16, 46, 24, 19, 33

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Dec 04 2009

			n=0: 2^53 = 9007199254740992, digit 0 duplicated at 9(00)7199254..
n=1: 2^40 = 1099511627776, digit 1 duplicated at 10995(11)627776..
n=2: 2^43 = 8796093022208, digit 2 duplicated even twice
n=3: 2^25 = 33554432, digit 3 duplicated at (33)554432
n=4: 2^18 = 262144
n=5: 2^16 = 65536
n=6: 2^46 = 70368744177664
n=7: 2^24 = 16777216
n=8: 2^19 = 524288
n=9: 2^33 = 8589934592

Helmut Kracke, Mathe-musische Knobelisken, Duemmler Bonn, 2. Auflage 1983

Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005

Cf. A000079, A018802.

Definition simplified, keyword:base,full added - R. J. Mathar, Dec 07 2009

A018857 Smallest power of 3 that begins with n.

Original entry on oeis.org

1, 27, 3, 4782969, 59049, 6561, 729, 81, 9, 10460353203, 1162261467, 129140163, 1350851717672992089, 14348907, 1594323, 16677181699666569, 177147, 1853020188851841, 19683, 205891132094649, 2187, 22876792454961

Offset: 1

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 1..563

Cf. A018802 (k=2), this sequence (k=3), A018859 (k=4), A018861 (k=5), A018863 (k=6), A018865 (k=7), A018867 (k=8), A018869 (k=9).

a(n) = 3^A018858(n). - Seiichi Manyama, Jan 29 2017

a(3^n) = 3^n for n >= 0. - Seiichi Manyama, Jan 29 2017

A018859 Smallest power of 4 that begins with n.

Original entry on oeis.org

1, 256, 302231454903657293676544, 4, 5070602400912917605986812821504, 64, 70368744177664, 81129638414606681695789005144064, 91343852333181432387730302044767688728495783936, 1024, 1125899906842624

Offset: 1

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 1..490

Cf. A018802 (k=2), A018857 (k=3), this sequence (k=4), A018861 (k=5), A018863 (k=6), A018865 (k=7), A018867 (k=8), A018869 (k=9).

a(n) = 4^A018860(n). - Michel Marcus, Feb 24 2016

a(4^n) = 4^n for n >= 0. - Seiichi Manyama, Jan 29 2017

A018863 Smallest power of 6 that begins with n.

Original entry on oeis.org

1, 216, 36, 46656, 50021738714629030177311081962496059484833406150976385567830453518336, 6, 7776, 80204967233062404407033075859456

Offset: 1

David W. Wilson

a(9) = 6^176, approximately 9.007827639*10^136. - Alan Frank, Jan 28 2011

a(10) = 6^9 = 10077696. - N. J. A. Sloane, Jan 28 2011

Seiichi Manyama, Table of n, a(n) for n = 1..279

Main sequence is A018864.

Cf. A018802 (k=2), A018857 (k=3), A018859 (k=4), A018861 (k=5), this sequence (k=6), A018865 (k=7), A018867 (k=8), A018869 (k=9).

a(n) = 6^A018864(n). - Seiichi Manyama, Jan 30 2017

a(6^n) = 6^n for n >= 0. - Seiichi Manyama, Jan 30 2017

A018865 Smallest power of 7 that begins with n.

Original entry on oeis.org

1, 2401, 343, 49, 5764801, 678223072849, 7, 823543, 96889010407, 107006904423598033356356300384937784807, 117649, 129934811447123020117172145698449, 13841287201, 1481113296616977741464105532513750734030421355207

Offset: 1

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 1..237

Cf. A018802 (k=2), A018857 (k=3), A018859 (k=4), A018861 (k=5), A018863 (k=6), this sequence (k=7), A018867 (k=8), A018869 (k=9).

a(n) = 7^A018866(n). - Seiichi Manyama, Jan 30 2017

a(7^n) = 7^n for n >= 0. - Seiichi Manyama, Jan 30 2017

One more term (a(14)) from Harvey P. Dale, Mar 04 2014

A018867 Smallest power of 8 that begins with n.

Original entry on oeis.org

1, 262144, 32768, 4096, 512, 64, 73786976294838206464, 8, 9223372036854775808, 1073741824, 1152921504606846976, 1237940039285380274899124224, 134217728, 144115188075855872, 154742504910672534362390528, 16777216

Offset: 1

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 1..342

Cf. A018802 (k=2), A018857 (k=3), A018859 (k=4), A018861 (k=5), A018863 (k=6), A018865 (k=7), this sequence (k=8), A018869 (k=9).

a(n) = 8^A018868(n). - Seiichi Manyama, Jan 29 2017

a(8^n) = 8^n for n >= 0. - Seiichi Manyama, Jan 29 2017

A171242 a(n) = k is the smallest exponent k such that at least 3 equal decimal digits "n n n" appear in the decimal representation of 2^k (n=0,1,...,9).

Original entry on oeis.org

242, 42, 43, 83, 44, 41, 157, 24, 39, 50

Offset: 0

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Dec 06 2009

			n=0: 2^242 = 7067388259113537318333190002971674063309935587502475832486424805170479104
n=1: 2^42 = 4398046511104
n=2: 2^43 = 8796093022208
n=3: 2^83 = 9671406556917033397649408
n=4: 2^44 = 17592186044416
n=5: 2^41 = 2199023255552
n=6: 2^157 = 182687704666362864775460604089535377456991567872
n=7: 2^24 = 16777216
n=8: 2^39 = 549755813888
n=9: 2^50 = 1125899906842624

E. I. Ignatjew, Mathematische Spielereien, Urania Verlag Leipzig-Jena-Berlin, 2. Auflage 1982
Helmut Kracke, Mathe-musische Knobelisken, Duemmler Bonn, 2. Auflage 1983

Cf. A000079, A018802, A171132.

Table[Module[{k=1},While[SequenceCount[IntegerDigits[2^k],{n,n,n}]<1,k++];k],{n,0,9}] (* Harvey P. Dale, Nov 28 2023 *)

Offset corrected by Alois P. Heinz, Nov 28 2023

cp's OEIS Frontend

A018856 2^a(n) is the smallest power of 2 beginning with n.

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A018869 Smallest power of 9 that begins with n.

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A018861 Smallest power of 5 that begins with n.

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A171132 The smallest exponent k such that the digit n appears at two adjacent places in the decimal representation of 2^k.

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A018857 Smallest power of 3 that begins with n.

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A018859 Smallest power of 4 that begins with n.

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A018863 Smallest power of 6 that begins with n.

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A018865 Smallest power of 7 that begins with n.

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A018867 Smallest power of 8 that begins with n.

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A171242 a(n) = k is the smallest exponent k such that at least 3 equal decimal digits "n n n" appear in the decimal representation of 2^k (n=0,1,...,9).

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