A050499 -id:A050499 - cp's OEIS frontend

A078607 Least positive integer x such that 2*x^n > (x+1)^n.

1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 40, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102

Offset: 0

Jon Perry, Dec 09 2002

nonn

Also, integer for which E(s) = s^n - Sum_{0 < k < s} k^n is maximal. It appears that a(n) + 2 is the least integer for which E(s) < 0. - M. F. Hasler, May 08 2020

			a(2) = 3 as 2^2 = 4, 3^2 = 9 and 4^2 = 16.
For n = 777451915729368, a(n) = 1121626023352384 = ceiling(n log 2), where n*log(2) = 1121626023352383.5 - 2.13*10^-17 and 2*floor(n log 2)^n / floor(1 + n log 2)^n = 1 - 3.2*10^-32. - _M. F. Hasler_, Nov 02 2013
a(2) is given by floor(1/(1-1/sqrt(2))). [From former A230748.]

Vincenzo Librandi, Table of n, a(n) for n = 0..20000

Cf. A224996 (the largest integer x that satisfies 2*x^n <= (x+1)^n).
Cf. A050499, A050500.
Cf. A078608, A078609. Equals A110882(n)-1 for n > 0.
Cf. A332097 (maximum of E(s), cf comments), also related to this:  A332101 (least k such that k^n <= sum of all smaller n-th powers), A030052 (least k such that k^n = sum of distinct n-th powers), A332065 (all k such that k^n is a sum of distinct n-th powers).

Table[SelectFirst[Range@ 120, 2 #^n > (# + 1)^n &], {n, 0, 71}] (* Michael De Vlieger, May 01 2016, Version 10 *)

for (n=2,50, x=2; while (2*x^n<=((x+1)^n),x++); print1(x","))

a(n)=1\(1-1/2^(1/n)) \\ Charles R Greathouse IV, Oct 31 2013

apply( A078607(n)=ceil(1/if(n>1,sqrtn(2,n)-1,!n+n/2)), [0..80]) \\ M. F. Hasler, May 08 2020

a(n) = ceiling(1/(2^(1/n)-1)) for n > 1. (For n = 1 resp. 0 this gives the integer 1 resp. infinity as argument of ceiling.) [Edited by M. F. Hasler, May 08 2020]
For most n, a(n) is the nearest integer to n/log(2), but there are exceptions, including n=777451915729368.
Following formulae merged in from former A230748, M. F. Hasler, May 14 2020:
a(n) = floor(1/(1-1/2^(1/n))).
a(n) = n/log(2) + O(1). - Charles R Greathouse IV, Oct 31 2013
a(n) = floor(1/(1-x)) with x^n = 1/2: f(n) = 1/(2^(1/n)-1) is never an integer for n > 1, whence floor(f(n)+1) = ceiling(f(n)) = a(n). - M. F. Hasler, Nov 02 2013, and Gabriel Conant, May 01 2016

Edited by Dean Hickerson, Dec 17 2002

Initial terms a(0) = 1 and a(1) = 2 added by M. F. Hasler, Nov 02 2013

A057834 Integer nearest to 10^n / log(10^n).

Original entry on oeis.org

4, 22, 145, 1086, 8686, 72382, 620421, 5428681, 48254942, 434294482, 3948131654, 36191206825, 334072678387, 3102103442166, 28952965460217, 271434051189532, 2554673422960305, 24127471216847324, 228576043106974646

Offset: 1

Robert G. Wilson v, Nov 08 2000

nonn

Legendre's Logarithmic Law: "In 1896, a full century after Adrienne Marie Legendre (1752 - 1833) guessed the approximate formula N/ln N for the number of primes up to N, Jacques Hadamard and Charles-Jacques de la Vallee-Poussin conclusively established it. They both lived for more than 50 years after producing their simultaneous but independent proofs." ... "In the meantime, Gauss and Riemann had made improved guesses expressed in terms of natural logarithms that we'll meet in Chapter 9."

John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1995, Pages 143 - 146.

Vladimir Pletser, Table of n, a(n) for n = 1..50
Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Table 1 p. 6.

[Round(10^n / Log(10^n)): n in [1..20]]; // Vincenzo Librandi, Jul 09 2015

Table[ Round[ N[ 10^n / Log[ 10^n ] ] ], {n, 1, 22} ]

a(n) = A050499(A011557(n)). - Henry Bottomley, Aug 10 2005

Corrected by Henry Bottomley, Aug 10 2005

A050500 Integer part of n/log(n).

Original entry on oeis.org

2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17

Offset: 2

N. J. A. Sloane, Dec 27 1999

nonn

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

See A050499 for more information.

Cf. A072626.

a[n_] := Floor[n / Log[n]]; Array[a, 100, 2] (* Amiram Eldar, Mar 17 2025 *)

A050501 a(n) = ceiling(n/log(n)).

Original entry on oeis.org

3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18

Offset: 2

N. J. A. Sloane, Dec 27 1999

nonn

Also, least k such that k*log(n) exceeds n. - Amarnath Murthy, Jun 17 2004

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

See A050499 for more information.

Do[k = 1; While[k*Log[n] <= n, k++ ]; Print[k], {n, 2, 100}] (* Ryan Propper, Sep 17 2005 *)
Array[ Ceiling[#/Log@#] &, 80, 2] (* Robert G. Wilson v, Jan 15 2017 *)

Edited by N. J. A. Sloane, Sep 15 2008 at the suggestion of R. J. Mathar

A053622 a(n) = round( n/log(n) - pi(n) ).

Original entry on oeis.org

2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, -1, -1, -2, -1, -1, -1, -2, -1, -1, -1, -1, -1, -1, -1, -2, -2, -2, -1, -1, -1, -2, -2, -1, -1, -2, -2, -3, -2, -2, -2, -3, -3, -2, -2, -2, -2, -3, -2, -2, -2, -2, -2, -3, -2, -3, -3, -3, -3, -2, -2, -3, -3, -3, -3, -3, -3

Offset: 2

Henry Bottomley, Mar 21 2000

sign

G. C. Greubel, Table of n, a(n) for n = 2..1000
C. K. Caldwell, How Many Primes Are There?

Cf. A000720, A050499, A052434, A052435.

[Round(n/Log(n) - #PrimesUpTo(n)): n in [2..80]]; // G. C. Greubel, May 17 2019

Table[Round[n/Log[n] - PrimePi[n]], {n,2,80}] (* G. C. Greubel, Apr 20 2017 *)

vector(80, n, n++; round(n/log(n) - primepi(n)) ) \\ G. C. Greubel, May 17 2019

[round(n/log(n) - prime_pi(n)) for n in (2..80)] # G. C. Greubel, May 17 2019

a(n) = A050499(n) - A000720(n).

A138194 Floor(2n/(3log(n))).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13

Offset: 2

Reinhard Zumkeller, Mar 04 2008

nonn

a(n) <= A000720(n) <= A138195(n), (Tschebyscheff, 1850).

Cf. A050500, A050499, A050501.

Table[Floor[(2n)/(3Log[n])],{n,2,100}] (* Harvey P. Dale, Jan 25 2025 *)

A138195 Floor(8n/(5log(n))).

Original entry on oeis.org

4, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28

Offset: 2

Reinhard Zumkeller, Mar 04 2008

nonn

A138194(n) <= A000720(n) <= a(n), (Tschebyscheff, 1850).

Cf. A050500, A050499, A050501.

Table[Floor[(8n)/(5Log[n])],{n,2,80}] (* Harvey P. Dale, Feb 14 2025 *)

A178805 Numbers k such that ||k/log(k)|| reaches a new minimum, where ||x|| is the distance from x to the nearest integer.

Original entry on oeis.org

2, 5, 9, 13, 17, 163, 53453, 110673, 715533, 1432276, 6517719, 11523158, 11985596, 24102781, 254977309, 451207448, 1219588338, 2048539023, 10066616717, 42116139191, 47657002570, 73831354169, 122478947521, 143949453227, 3040705645816, 3152420311977, 5624690531099, 14964977749017, 25999244327633, 92799025313425, 164330745650026, 604329910739082

Offset: 1

T. D. Noe, Jun 16 2010

nonn

The first 9 terms are from Kevin O'Bryant in a Math OverFlow comment. Closely related to A178806.

Added 15 more terms from Noam D. Elkies (second MathOverflow link). - T. D. Noe, Feb 12 2013

MathOverflow, Why Is 163/ln(163) a Near-Integer?
MathOverflow, When is n/ln(n) close to an integer?

Cf. A050499 (nearest integer to n/log(n)), A178806.

mn=Infinity; n=2; Table[While[r=N[n/Log[n]]; diff=Abs[r-Round[r]]; diff>=mn, n++ ]; mn=diff; Print[{n,mn}]; n, {9}]

A053621 Nearest integer to n/(log(n)-1).

Original entry on oeis.org

-1, -7, 30, 10, 8, 8, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22

Offset: 1

Henry Bottomley, Mar 21 2000

sign

n/(log(n)-1) is a better approximation than n/log(n) to pi(n) the number of primes <= n, though worse than the logarithmic integral or the Riemann prime number formula.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. K. Caldwell, How Many Primes Are There?
Pierre Dusart, Estimates of Some Functions Over Primes without R.H., arXiv:1002.0442 [math.NT], 2010.

Cf. A000720, A047784, A050499, A053620.

a053621 = round . (\x -> x / (log x - 1)) . fromIntegral
-- Reinhard Zumkeller, Apr 30 2014

[Round(n/(Log(n)-1)): n in [1..80]]; // G. C. Greubel, May 17 2019

Table[Round[n/(Log[n]-1)], {n,1,80}] (* G. C. Greubel, May 17 2019 *)

vector(80, n, round(n/(log(n)-1))) \\ G. C. Greubel, May 17 2019

[round(n/(log(n)-1)) for n in (1..80)] # G. C. Greubel, May 17 2019

A178806 Numbers n such that ||n/log(n)|| * log(n) reaches a new minimum, where ||x|| is the distance from x to the nearest integer.

Original entry on oeis.org

2, 17, 163, 715533, 1432276, 6517719, 11523158, 11985596, 24102781, 254977309, 451207448, 1219588338, 2048539023, 10066616717, 42116139191, 47657002570, 73831354169, 122478947521, 143949453227, 3152420311977, 5624690531099, 14964977749017, 25999244327633, 92799025313425, 164330745650026, 604329910739082

Offset: 1

T. D. Noe, Jun 16 2010

nonn

The first 9 terms are from Will Jagy in a MathOverflow answer. Closely related to A178805.

Added 14 more terms from Noam D. Elkies (second MathOverflow link). - T. D. Noe, Feb 12 2013

MathOverflow, When is n/ln(n) close to an integer?
MathOverflow, When is n/ln(n) close to an integer?

Cf. A050499 (nearest integer to n/log(n)).

mn=Infinity; n=2; Table[While[r=N[n/Log[n]]; diff=Log[n] Abs[r-Round[r]]; diff>=mn, n++ ]; mn=diff; Print[{n,mn}]; n, {9}]

cp's OEIS Frontend

A078607 Least positive integer x such that 2*x^n > (x+1)^n.

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A057834 Integer nearest to 10^n / log(10^n).

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A050500 Integer part of n/log(n).

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A050501 a(n) = ceiling(n/log(n)).

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A053622 a(n) = round( n/log(n) - pi(n) ).

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A138194 Floor(2*n/(3*log(n))).

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A138195 Floor(8*n/(5*log(n))).

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A178805 Numbers k such that ||k/log(k)|| reaches a new minimum, where ||x|| is the distance from x to the nearest integer.

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A138194 Floor(2n/(3log(n))).

A138195 Floor(8n/(5log(n))).