A066343 -id:A066343 - cp's OEIS frontend

A067497 Smallest k for which 2^k is n+1 decimal digits long, and equivalently numbers k such that 1 is the first digit of 2^k.

0, 4, 7, 10, 14, 17, 20, 24, 27, 30, 34, 37, 40, 44, 47, 50, 54, 57, 60, 64, 67, 70, 74, 77, 80, 84, 87, 90, 94, 97, 100, 103, 107, 110, 113, 117, 120, 123, 127, 130, 133, 137, 140, 143, 147, 150, 153, 157, 160, 163, 167, 170, 173, 177, 180, 183, 187, 190, 193, 196

Offset: 0

Benoit Cloitre, Feb 22 2002

The asymptotic density of this sequence is log_10(2) = 0.301029... (A007524). - Amiram Eldar, Jan 27 2021

Muniru A Asiru, Table of n, a(n) for n = 0..5000

Cf. A000079, A007524, A066343, A067480, A067488, A129344.

Filtered([0..200],n->ListOfDigits(2^n)[1]=1); # Muniru A Asiru, Oct 22 2018

a[n_] := Block[{k = 0}, While[ Floor[Log[10, 2^k] + 1] < n, k++ ]; k]; Table[ a[n], {n, 1, 61}]
Table[Ceiling[n*Log[2, 10]], {n, 0, 59}] (* Jean-François Alcover, Jan 29 2014, after Vladeta Jovovic *)

for(n=0,500, if(floor(2^n/10^(floor(n*log(2)/log(10))))==1,print1(n,", ")))

a(n) = ceil(n*log(10)/log(2)); \\ Michel Marcus, May 13 2017

def A067497(n): return (10**n-1).bit_length() # Chai Wah Wu, Apr 02 2023

[ceil(n*log(10)/log(2)) for n in range(0, 60)] # Stefano Spezia, Aug 31 2024

a(n) = ceiling(n*log_2(10)). - Vladeta Jovovic, Jun 20 2002

a(n) = log_2(A067488(n+1)). - Charles L. Hohn, Jun 09 2024

Additional comments from Lekraj Beedassy, Jun 20 2002 and from Rick Shephard, Jun 27 2002

A123384 Number of bits in binary expansion of 10^n.

Original entry on oeis.org

1, 4, 7, 10, 14, 17, 20, 24, 27, 30, 34, 37, 40, 44, 47, 50, 54, 57, 60, 64, 67, 70, 74, 77, 80, 84, 87, 90, 94, 97, 100, 103, 107, 110, 113, 117, 120, 123, 127, 130, 133, 137, 140, 143, 147, 150, 153, 157, 160, 163, 167, 170, 173, 177, 180, 183, 187, 190, 193, 196

Offset: 0

Andrew Caldwell (spongebobpj(AT)yahoo.com), Nov 09 2006

Number of powers of 2 less than or equal to 10^n. - Peter Munn, Nov 13 2019

			a(3)=10 because 10^3 = 1111101000_2.
10^1 = 10 = 1010_2 has 4 digits.

Cf. A000079, A007524, A011557, A066343, A067497.

Row 1 of A253635.

A007524 := log[10](2.0) ; for n from 0 to 40 do printf("%d,", 1+floor(n/A007524)) ; od: # R. J. Mathar, Nov 12 2006
a:=n->nops(convert(10^n,base,2)): seq(a(n),n=0..70); # Emeric Deutsch, Mar 26 2007

a[n_]:=1 + Floor[n/Log10[2]]; Array[a,60,0] (* Stefano Spezia, Aug 31 2024 *)

a(n) = 1 + floor(n/A007524) = 1 + floor(n/log_10(2)). - R. J. Mathar, Nov 12 2006
a(n) = 1 + A066343(n). - R. J. Mathar, Mar 02 2007
a(n) = A067497(n) for n >= 1. - Georg Fischer, Nov 02 2018

More terms from Emeric Deutsch, Mar 26 2007

A253635 Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 4, 1, 7, 7, 1, 9, 20, 10, 1, 10, 34, 40, 14, 1, 10, 46, 86, 67, 17, 1, 10, 55, 141, 175, 101, 20, 1, 10, 62, 192, 338, 313, 142, 24, 1, 10, 67, 242, 522, 694, 507, 190, 27, 1, 10, 72, 287, 733, 1197, 1273, 768, 244, 30

Offset: 1

L. Edson Jeffery, Jan 07 2015

Or a(n,k) = the number of positive integers less than or equal to 10^k that are divisible by no prime exceeding prime(n).

			Array begins:
{1,  4,  7,  10,   14,   17,    20,    24,    27,     30, ...}
{1,  7, 20,  40,   67,  101,   142,   190,   244,    306, ...}
{1,  9, 34,  86,  175,  313,   507,   768,  1105,   1530, ...}
{1, 10, 46, 141,  338,  694,  1273,  2155,  3427,   5194, ...}
{1, 10, 55, 192,  522, 1197,  2432,  4520,  7838,  12867, ...}
{1, 10, 62, 242,  733, 1848,  4106,  8289, 15519,  27365, ...}
{1, 10, 67, 287,  945, 2579,  6179, 13389, 26809,  50351, ...}
{1, 10, 72, 331, 1169, 3419,  8751, 20198, 42950,  85411, ...}
{1, 10, 76, 369, 1385, 4298, 11654, 28434, 63768, 133440, ...}
{1, 10, 79, 402, 1581, 5158, 14697, 37627, 88415, 193571, ...}

Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108276, A108277 (rows 1-9).

Cf. A011557, A066343, A253572, A253573.

r = 10; y[1] = t = Table[2^j, {j, 0, 39}]; max = 10^13; len = 10^10; prev = 0; For[n = 2, n <= r, n++, next = 0; For[k = 1, k <= 43, k++, If[Prime[n]^k < max, t = Union[t, Prime[n]*t]; s = FirstPosition[t, v_ /; v > len, 0]; t = Take[t, s[[1]] - 1]; If[t[[-1]] > len, t = Delete[t, -1]]; next = Length[t]; If[next == prev, Break, prev = next], Break]]; y[n] = t]; b[i_, j_] := FirstPosition[y[i], v_ /; v > 10^j][[1]]; a253635[n_, j_] := If[IntegerQ[b[n, j]], b[n, j] - 1, 0]; Flatten[Table[a253635[n - j, j], {n, r}, {j, 0, n - 1}]] (* array antidiagonals flattened *)

A074116 Largest n-digit power of 2.

Original entry on oeis.org

8, 64, 512, 8192, 65536, 524288, 8388608, 67108864, 536870912, 8589934592, 68719476736, 549755813888, 8796093022208, 70368744177664, 562949953421312, 9007199254740992, 72057594037927936, 576460752303423488, 9223372036854775808, 73786976294838206464, 590295810358705651712

Offset: 1

Amarnath Murthy, Aug 27 2002

The exponents are given in A066343. - Evgeny Kapun, Jan 16 2017

An equivalent definition (which was formerly the definition of A074113): "Smallest n-digit number of the form p^a*q^b... with the maximum value of a+b+.... where p, q etc. are primes. If a,b,c,... are the indices in the signature prime factorization then a+b+c ... is a maximum." That this is the same sequence follows from the inequality p^a*q^b... >= 2^(a+b+...) and the fact that there always exists a power of 2 between two consecutive powers of 10.

Evgeny Kapun, Table of n, a(n) for n = 1..1000

Cf. A066343, A067488, A074113, A074114, A074117, A074118.

Last[#]&/@(With[{l2=2^Range[80]},Table[Select[l2,IntegerLength[#] == n&], {n,22}]]) (* Harvey P. Dale, Jul 17 2011 *)

a(n) = 2^A066343(n).

Edited by R. J. Mathar, Feb 13 2008, Max Alekseyev, Mar 10 2009, Harvey P. Dale, Jul 17 2011, Evgeny Kapun, Jan 16 2017, and N. J. A. Sloane, Jan 18 2017

A100752 a(n) is the number of positive integers <= 10^n that are divisible by no prime exceeding 3.

Original entry on oeis.org

1, 7, 20, 40, 67, 101, 142, 190, 244, 306, 376, 452, 534, 624, 720, 824, 935, 1052, 1178, 1309, 1447, 1593, 1745, 1905, 2071, 2244, 2424, 2611, 2806, 3006, 3214, 3429, 3652, 3881, 4117, 4360, 4610, 4866, 5131, 5401, 5679, 5964, 6255, 6553, 6859, 7172, 7491

Offset: 0

Robert G. Wilson v, May 27 2005

nonn

A good approximation seems to be ceiling(log(10^n)*log(6*10^n)/(log(3)*log(4))). - Horst H. Manninger, Oct 29 2022

			a(1) = 7 as there are 7 3-smooth numbers less than 10^1 = 10; they are 1, 2, 3, 4, 6, 8, 9. - _David A. Corneth_, Nov 14 2019

David A. Corneth, Table of n, a(n) for n = 0..1999

Cf. A003586, A011557, A071521.
Cf. A066343, A106598, A106600, A107352, A106629.
Row 2 of A253635.

f[n_] := Sum[ Floor@ Log[2, n/3^i] + 1, {i, 0, Log[3, n]}]; Table[ f[10^n], {n, 0, 46}] (* Robert G. Wilson v, Nov 07 2012 *)

from sympy import integer_log
def A100752(n): return sum((10**n//3**i).bit_length() for i in range(integer_log(10**n,3)[0]+1)) # Chai Wah Wu, Oct 23 2024

a(n) = A071521(10^n). - Chai Wah Wu, Oct 23 2024

A066344 Beatty sequence for log_5(10).

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 12, 14, 15, 17, 18, 20, 21, 22, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 60, 61, 62, 64, 65, 67, 68, 70, 71, 72, 74, 75, 77, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 94, 95, 97, 98, 100

Offset: 1

Vladeta Jovovic, Dec 15 2001

Beatty complement of A066343. - Jianing Song, Jan 27 2019

Harry J. Smith, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A066343, A154156 (log_5(10)).

Floor[Range[100]*Log[5, 10]] (* Paolo Xausa, Jul 08 2024 *)

{ l=log(10)/log(5); for (n=1, 1000, a=floor(n*l); write("b066344.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 11 2010

from sympy import integer_log
def A066344(n): return integer_log(1<Chai Wah Wu, Sep 08 2024

a(n) = floor(n*log_5(10)).

A129344 a(n) is the number of powers of 2 that have n decimal digits.

Original entry on oeis.org

4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3

Offset: 1

Tanya Khovanova, May 28 2007

Ignoring the first term, first differences of A066343. - Andrew Woods, Jun 10 2013

			a(1) is 4 because there are 4 one-digit powers of 2: 1, 2, 4, 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000

First differences of A067497.

Cf. A000079, A020862, A066343.

Table[Transpose[ Select[Table[{n, 2^n}, {n, 0, 310}], IntegerDigits[ #[[2]]][[1]] == 1 &]][[1]][[k]] - Transpose[ Select[Table[{n, 2^n}, {n, 0, 310}], IntegerDigits[ #[[2]]][[1]] == 1 &]][[1]][[k - 1]], {k, 2, 94}]
Join[{4}, Differences @ Table[Floor[n*Log2[10]], {n, 100}]] (* Amiram Eldar, Apr 09 2021 *)

a(n) = my(k=0, i=0); while(#Str(2^k)!=n, k++); while(#Str(2^k)==n, i++; k++); i \\ Felix Fröhlich, Jan 19 2016

def A129344(n): return -(m:=5**(n-1)).bit_length()+(5*m).bit_length()+1 if n>1 else 4 # Chai Wah Wu, Sep 08 2024

For n>1, a(n) = floor(n*L)-floor((n-1)*L) where L = log(10)/log(2). - Andrew Woods, Jun 10 2013

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log_2(10) (A020862). - Amiram Eldar, Apr 09 2021

A054965 Beatty sequence for log_3(10), i.e., for 1/log_10(3); so largest exponent of 3 which produces an n-digit decimal number.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 132, 134, 136

Offset: 1

Henry Bottomley, Dec 13 2002

			log_10(3) = 0.477121... so a(11) = floor(11/0.477121...) = floor(23.0549...) = 23; 3^23 = 94143178827 is the largest 11 decimal digit power of 3.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A062153, A066343, A074118, A152566.

Floor[Range[100]*Log[3, 10]] (* Paolo Xausa, Jul 11 2024 *)

a(n) = n*log(10)\log(3); \\ Michel Marcus, Aug 03 2017

a(n) = floor(n/log_10(3)) = log_3(A074118(n)) = A062153(A074118(n)).

A273158 Number of concatenations nm consisting of n followed by a positive integer m (not a multiple of 10) that are divisible by m.

Original entry on oeis.org

5, 6, 12, 8, 8, 14, 14, 9, 21, 9, 14, 17, 16, 17, 19, 11, 16, 25, 16, 11, 32, 17, 16, 20, 11, 18, 33, 20, 17, 21, 17, 12, 33, 18, 20, 29, 17, 18, 35, 12, 17, 37, 17, 20, 31, 18, 17, 23, 26, 12

Offset: 1

Reiner Moewald, May 16 2016

			a(1)=5 since 1|11, 2|12, 5|15, 25|125, 125|1125.

Reiner Moewald, Table of n, a(n) for n = 1..98

Cf. A066343.

# for N < 100
import math
for N in range (1, 100):
   ANZ = 0
   for M in range(1, 195312500):
      Z = int(str(int(N)) + str(int(M)))
      if ((Z % M == 0) and (M % 10 > 0)):
         ANZ = ANZ + 1
   print(N, ANZ)

m <= n * 5^a(2 + floor(log_10(n))) with a(n) from A066343.

cp's OEIS Frontend

A067497 Smallest k for which 2^k is n+1 decimal digits long, and equivalently numbers k such that 1 is the first digit of 2^k.

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A123384 Number of bits in binary expansion of 10^n.

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A253635 Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.

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A074116 Largest n-digit power of 2.

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A100752 a(n) is the number of positive integers <= 10^n that are divisible by no prime exceeding 3.

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A066344 Beatty sequence for log_5(10).

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A129344 a(n) is the number of powers of 2 that have n decimal digits.

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