A050500 -id:A050500 - cp's OEIS frontend

A050499 Nearest integer to n/log(n).

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

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

Original entry on oeis.org

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

A072626 Parity of floor(n/log(n)).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0

Offset: 2

Labos Elemer, Jun 28 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A000035, A000720, A050500, A071986.

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

a(n) = floor(n / log(n)) mod 2.

a(n) = A000035(A050500(n)). - Amiram Eldar, Mar 17 2025

Offset corrected by Sean A. Irvine, Oct 15 2024

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 *)

A141602 Integer part of 2^n/log(2^n).

Original entry on oeis.org

2, 2, 3, 5, 9, 15, 26, 46, 82, 147, 268, 492, 909, 1688, 3151, 5909, 11123, 21010, 39809, 75638, 144073, 275050, 526182, 1008516, 1936352, 3723754, 7171675, 13831089, 26708310, 51636066, 99940774, 193635250, 375535031, 728979766, 1416303547

Offset: 1

Cino Hilliard, Aug 21 2008

nonn

2^n/log(2^n) is an approximation to the number of primes < 2^n.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000799, A007525, A050500, A185192.

A141602:= func< n | Floor(2^n/(n*Log(2))) >;
[A141602(n): n in [1..40]]; // G. C. Greubel, Sep 21 2024

Floor[2^#/Log[2^#]]&/@Range[40] (* Harvey P. Dale, Mar 11 2013 *)

g(n) = for(x=1,n,y=floor(2^x/log(2^x));print1(y","))

a(n) = 2^n\log(2^n); \\ Michel Marcus, Feb 24 2021

def A141602(n): return int(2^n/(n*log(2)))
[A141602(n) for n in range(1,41)] # G. C. Greubel, Sep 21 2024

a(n) = A050500(2^n) = floor(2^n*A007525/n) >= A000799(n). - R. J. Mathar, Jan 05 2009

A316350 Positive integers x that are x/log(x) smooth, that is, if a prime p divides x, then p <= x/log(x).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 77, 78, 80, 81, 84, 85, 88, 90, 91, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 119, 120

Offset: 1

Richard Locke Peterson, Jun 29 2018

nonn

This sequence is a monoid under multiplication, since if x and y are terms in the sequence and p < x/log(x), then p < xy/log(xy). However, if a term in the sequence is multiplied by a number outside the sequence, the result need not be in the sequence.

			1 is in the sequence because no primes divide 1, 2 is in the sequence since 2 divides 2 and 2 < 2/log(2) ~ 2.9, but 10 is not in the sequence since 5 divides 10 and 5 is not less than 10/log(10) ~ 4.34.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A050500.

filter:= n -> is(max(numtheory:-factorset(n))Robert Israel, Oct 21 2021

ok[n_] := AllTrue[First /@ FactorInteger[n], # Log[n] <= n &]; Select[ Range[120], ok] (* Giovanni Resta, Jun 30 2018 *)

isok(n) = my(f=factor(n)); for (k=1, #f~, if (f[k,1] >= n/log(n), return(0))); return (1); \\ Michel Marcus, Jul 02 2018

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A050499 Nearest integer to n/log(n).

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

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A072626 Parity of floor(n/log(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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A141602 Integer part of 2^n/log(2^n).

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A316350 Positive integers x that are x/log(x) smooth, that is, if a prime p divides x, then p <= x/log(x).

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

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