A002285 -id:A002285 - cp's OEIS frontend

A220261 Decimal expansion of (1/e) * log_10(e), where 1/e = A068985, log_10(e) = A002285.

1, 5, 9, 7, 6, 8, 0, 1, 1, 3, 0, 6, 4, 0, 9, 3, 5, 2, 6, 7, 2, 1, 4, 4, 3, 2, 7, 7, 1, 3, 3, 1, 9, 3, 9, 0, 8, 5, 8, 7, 2, 4, 0, 5, 1, 6, 1, 5, 3, 4, 5, 1, 8, 8, 1, 4, 0, 9, 0, 9, 2, 6, 3, 8, 0, 7, 5, 6, 6, 6, 4, 7, 1, 6, 1, 0, 1, 1, 2, 0, 7, 9, 4, 9, 5, 8

Offset: 0

Jaroslav Krizek, Dec 08 2012

Maximal value of function x * log_10(x) for x < 0 in x = -1/e.
-0,159768… = minimal value of function x * log_10(x) for x > 0 in x = 1/e.
Decimal expansion of 1 / (e * log(10)), where e = A001113, log(10) = A002392.

			0.15976801130640935267214...

Cf. A001113, A068985, A002285.

RealDigits[1/E Log10[E],10,120][[1]] (* Harvey P. Dale, Aug 11 2019 *)

A050499 Nearest integer to n/log(n).

Original entry on oeis.org

3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17

Offset: 2

N. J. A. Sloane, Dec 27 1999

nonn

The prime number theorem states that the number of primes <= x is asymptotic to x/log(x).
n/log(n) = n*A002285/log_10(n). [Eric Desbiaux, Jun 27 2009]
Similar to floor(1/(1-x)) where x^n=1/n. - Jon Perry, Oct 29 2013

Cf. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 6.

T. D. Noe, Table of n, a(n) for n=2..10000

Cf. A000720, A050500, A050501.

for (i=1;i<100;i++) {
x=Math.pow(1/i,1/i);
document.write(Math.floor(1/(1-x))+", ");
}

Table[Round[n/Log[n]],{n,2,80}] (* Harvey P. Dale, Nov 03 2013 *)

a(n) = round(n/log(n)); \\ Michel Marcus, Jan 24 2025

A059549 Beatty sequence for 1 + 1/log(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, 43, 44, 45, 47, 48, 50, 51, 53, 54, 55, 57, 58, 60, 61, 63, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 78, 80, 81, 83, 84, 86, 87, 88, 90, 91, 93, 94, 96, 97, 98, 100

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059550.

Cf. A002285.

Floor[Range[100]*(1 + Log10[E])] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=1 + 1/log(10); for (n = 1, 2000, write("b059549.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*(1 + 1/log(10))). - Michel Marcus, Jan 04 2015

A099260 Number of decimal digits in (10^n)-th prime number.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 0

Rick L. Shepherd, Oct 10 2004

As lim {n->oo} p_n/(n log n) = 1 is equivalent to the prime number theorem, a good first approximation (without having done any detailed analysis) should be a(n)=floor(log_10((10^n)*log(10^n))), which correctly generates all the first 22 terms and predicts that the sequence will continue 24,25,...,43,44,46,47,...,435,436,438,439,...,4344,4345,4347,4348,...,4503,4504 through the first 4500 terms (with only 5,45,437,4346 not appearing - compare with the digits of log_10(e) in A002285).

Many terms of this sequence can be determined exactly using Dusart's bounds. The first missing terms are 5, 44, 435, 4344, 43430, 434295, 4342946, 43429449, 434294483, 4342944820, ....

			a(4) = 6 because A006988(4) = prime(10^4) = 104729 has six decimal digits.

Pierre Dusart, Estimates of Some Functions Over Primes without R.H.

Cf. A006988 ((10^n)-th prime), A006880 (pi(10^n)), A099261 (bit lengths).

Table[IntegerLength[Prime[10^n]],{n,0,75}] (* Harvey P. Dale, Dec 11 2020 *)

a(n)=if(n<3,return(n+1));my(l=n*log(10),ll=log(l),lb=ceil(log(l+ll-1+(ll-2.2)/l)/log(10)),ub=ceil(log(l+ll-1+(ll-2)/l)/log(10)));if(lb==ub,n+lb,error("Cannot determine a("n")"))

Extension, comment, link, and Pari program from Charles R Greathouse IV, Aug 03 2010

A114467 Number of decimal digits in the numerator of the 10^n-th harmonic number.

Original entry on oeis.org

1, 4, 41, 434, 4346, 43451, 434111, 4342303, 43428680

Offset: 0

Eric W. Weisstein, Nov 29 2005

See A114468 for denominators.

Eric Weisstein's link conjectures that for both this sequence and A114468, a(n) ~ (log_10(e) = A002285)*10^n. - Natalia L. Skirrow, Jun 22 2023

J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number

Cf. A001008, A002285, A002805, A114468.

Table[IntegerLength[Numerator[HarmonicNumber[10^n]]],{n,0,8}](* Harvey P. Dale, May 24 2019 *)

from gmpy2 import digits, mpq
def a(n): return len(digits(sum(mpq(1, n) for n in range(1, 10**n+1)).numerator))
print([a(n) for n in range(6)]) # Michael S. Branicky, Jun 22 2023

Edited by Charles R Greathouse IV, Aug 05 2010

A059184 Engel expansion of 1/log(10) = 0.434294....

Original entry on oeis.org

3, 4, 5, 18, 27, 37, 415, 445, 1812, 2475, 3928, 6707, 14673, 65863, 89033, 265841, 1293955, 1525697, 2541166, 7906280, 107955268, 154190828, 887303031, 1767107652, 3068165143, 3209500563, 92762706640, 147991352023

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A002285 (1/log(10)).

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[1/Log[10], 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)

A130787 Decimal expansion of the square of Pi*log_10(e).

Original entry on oeis.org

1, 8, 6, 1, 5, 2, 2, 8, 3, 4, 9, 2, 2, 7, 5, 7, 4, 6, 4, 6, 8, 3, 2, 1, 5, 2, 1, 8, 7, 6, 2, 9, 8, 2, 3, 6, 7, 6, 1, 2, 9, 3, 7, 2, 8, 5, 8, 1, 3, 6, 9, 6, 5, 4, 4, 4, 2, 2, 0, 8, 2, 6, 3, 3, 2, 1, 1, 5, 2, 3, 8, 6, 2, 0, 1, 9, 2, 7, 2, 3, 6, 7, 1, 1, 5, 2, 6, 0, 6, 1, 6, 3, 1, 3, 7, 7, 5, 4, 4, 9, 8, 1, 5, 5, 0

Offset: 1

R. J. Mathar, Jul 15 2007

			1.86152283492275746468321521876298236761293728581369654442208263321...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
L. M. Milne-Thomson in M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, 17.3.20.

Cf. A002388, A002285.

Maple
```
evalf((Pi/log(10) )^2);
```

RealDigits[ (Pi/Log@10)^2, 10, 111][[1]] (* Robert G. Wilson v, Jul 19 2007 *)

Equals A002388*A002285^2 = (Pi/log(10))^2. - Robert G. Wilson v, Jul 19 2007

More terms from Robert G. Wilson v, Jul 19 2007

A220260 Decimal expansion of e / log_10(e), where e = A001113.

Original entry on oeis.org

6, 2, 5, 9, 0, 7, 5, 2, 1, 6, 7, 6, 6, 3, 9, 5, 2, 1, 1, 0, 1, 8, 4, 7, 7, 3, 1, 1, 6, 7, 7, 7, 2, 7, 2, 8, 0, 5, 4, 6, 4, 5, 9, 9, 5, 4, 2, 5, 9, 4, 6, 9, 4, 2, 4, 2, 7, 7, 6, 9, 8, 1, 6, 9, 8, 5, 8, 4, 5, 3, 9, 9, 1, 7, 4, 4, 2, 2, 9, 9, 2, 4, 2, 8, 4, 1, 2

Offset: 1

Jaroslav Krizek, Dec 08 2012

Minimum value of the function x / log_10(x) for x > 1, obtained at x = e.

			6.2590752167663952110184773116777272805464599542594694...

Cf. A001113, A002285, A002392.

RealDigits[E/Log[10,E],10,120][[1]] (* Harvey P. Dale, Dec 23 2012 *)

Equals the ratio A001113 / A002285 and also the product A001113*A002392.

A275238 a(n) = n*(10^floor(log_10(n)+1) + 1) + (-1)^n.

Original entry on oeis.org

1, 10, 23, 32, 45, 54, 67, 76, 89, 98, 1011, 1110, 1213, 1312, 1415, 1514, 1617, 1716, 1819, 1918, 2021, 2120, 2223, 2322, 2425, 2524, 2627, 2726, 2829, 2928, 3031, 3130, 3233, 3332, 3435, 3534, 3637, 3736, 3839, 3938, 4041, 4140, 4243, 4342, 4445, 4544, 4647, 4746, 4849, 4948, 5051, 5150, 5253, 5352, 5455, 5554

Offset: 0

Ilya Gutkovskiy, Jul 21 2016

Concatenation of n with n+(-1)^n (A004442).
Subsequence of A248378.
Primes in this sequence: 23, 67, 89, 1213, 3637, 4243, 5051, 5657, 6263, 6869, 7879, 8081, 9091, 9293, 9697, 102103, ... (A030458).
Numbers n such that a(n) is prime: 2, 6, 8, 12, 36, 42, 50, 56, 62, 68, 78, 80, 90, 92, 96, 102, 108, 120, 126, 138, ... (A030457).

			a(0) =  0 + 1 = 1;
a(1) = 11 - 1 = 10;
a(2) = 22 + 1 = 23;
a(3) = 33 - 1 = 32;
a(4) = 44 + 1 = 45;
a(5) = 55 - 1 = 54, etc.
or
a(0) =  1 -> concatenation of 0 with 0 + (-1)^0 = 1;
a(1) = 10 -> concatenation of 1 with 1 + (-1)^1 = 0;
a(2) = 23 -> concatenation of 2 with 2 + (-1)^2 = 3;
a(3) = 32 -> concatenation of 3 with 3 + (-1)^3 = 2;
a(4) = 45 -> concatenation of 4 with 4 + (-1)^4 = 5;
a(5) = 54 -> concatenation of 5 with 5 + (-1)^5 = 4, etc.
........................................................
a(2k) = 1, 23, 45, 67, 89, 1011, 1213, 1415, 1617, 1819, ...

Cf. A001704, A002285, A004442, A020338, A030457, A030458, A030656, A033999, A064834, A248378.

Table[n (10^Floor[Log[10, n] + 1] + 1) + (-1)^n, {n, 0, 55}]

a(n) = if(n, n*(10^(logint(n,10)+1) + 1) + (-1)^n, 1) \\ Charles R Greathouse IV, Jul 21 2016

a(n) = A020338(n) + A033999(n).
a(2k) = A030656(k).
A064834(a(n)) > 0, for n > 0.
a(n) ~ 10*n*10^floor(c*log(n)), where c = 1/log(10) = 0.4342944819... = A002285.

A282152 Decimal expansion of 20/log(10).

Original entry on oeis.org

8, 6, 8, 5, 8, 8, 9, 6, 3, 8, 0, 6, 5, 0, 3, 6, 5, 5, 3, 0, 2, 2, 5, 7, 8, 3, 7, 8, 3, 3, 2, 1, 0, 1, 6, 4, 5, 8, 8, 7, 9, 4, 0, 1, 1, 6, 0, 7, 3, 3, 3, 1, 3, 2, 2, 2, 8, 9, 0, 7, 5, 6, 6, 3, 3, 1, 7, 2, 9, 2, 9, 8, 4, 1, 7, 7, 4, 1, 5, 4, 9, 4, 5, 8, 4, 4, 9, 8, 9, 8, 6, 7, 6, 8, 6, 3, 4, 9, 6, 6, 3, 7, 4, 1, 2

Offset: 1

Arkadiusz Wesolowski, Feb 20 2017

			1 neper = 8.685889638 dB.
8.685889638065036553022578378332101645887940116073331322289075663317292984177...

Herman Medwin, "Sounds in the Sea: From Ocean Acoustics to Acoustical", Cambridge University Press, 2005, pp. 68-70.

Wikipedia, Neper
Index entries for transcendental numbers

Cf. A002392.

SetDefaultRealField(RealField(105)); n:=20/Log(10); Reverse(Intseq(Floor(10^104*n)));

RealDigits[20/Log[10], 10, 111][[1]] (* Robert G. Wilson v, Feb 20 2017 *)

PARI
```
20/log(10)
```

Equals 20*A002285 = 20/A002392.

cp's OEIS Frontend

A220261 Decimal expansion of (1/e) * log_10(e), where 1/e = A068985, log_10(e) = A002285.

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A050499 Nearest integer to n/log(n).

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A059549 Beatty sequence for 1 + 1/log(10).

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A099260 Number of decimal digits in (10^n)-th prime number.

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A114467 Number of decimal digits in the numerator of the 10^n-th harmonic number.

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A059184 Engel expansion of 1/log(10) = 0.434294....

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A130787 Decimal expansion of the square of Pi*log_10(e).

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Maple

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A220260 Decimal expansion of e / log_10(e), where e = A001113.

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