A050501 -id:A050501 - cp's OEIS frontend

A096056 Duplicate of A050501.

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

Offset: 2

dead

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

A276523 Partition an n X n square into multiple non-congruent integer-sided rectangles. a(n) is the least possible difference between the largest and smallest area.

Original entry on oeis.org

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

Offset: 3

Ed Pegg Jr, Nov 15 2016

Developed as the Mondrian Art Puzzle.
The rectangles can be similar, though. - Daniel Forgues, Nov 22 2016
That is, there can be a 1x2 rectangle and a 2x4 rectangle (these are similar), but there can't be two 1x2 rectangles (these are congruent). - Michael B. Porter, Oct 13 2018
Upper bounds for a(n) are n if n is odd, and min(2*n, 4 * a(n/2)) if n is even. - Roderick MacPhee, Nov 28 2016
An upper bound seems to be ceiling(n/log(n))+3, or A050501+3. See A278970. Holds to at least a(96). - Ed Pegg Jr, Dec 02 2016
Best known values for a(66)-a(96) as follows: 16, 18, 19, 18, 19, 18, 20, 20, 20, 20, 19, 20, 21, 21, 20, 21, 20, 20, 21, 22, 18, 22, 20, 22, 24, 23, 22, 22, 24, 24, 24. - (shortened by Ruud H.G. van Tol, Oct 25 2024)

			A size-11 square can be divided into 3 X 4, 2 X 6, 2 X 7, 3 X 5, 4 X 4, 2 X 8, 2 X 9, and 3 X 6 rectangles. 18 - 12 = 6, the minimal area range.
The 14 X 14 square can be divided into non-congruent rectangles of area 30 to 36:
  aaaaaaaaaabbbb
  aaaaaaaaaabbbb
  aaaaaaaaaabbbb
  cccdddddddbbbb
  cccdddddddbbbb
  cccdddddddbbbb
  cccdddddddbbbb
  cccdddddddbbbb
  ccceeeeeffffff
  ccceeeeeffffff
  ccceeeeeffffff
  ccceeeeeffffff
  ccceeeeeffffff
  ccceeeeeffffff

Michel Gaillard, Optimal tilings for n=58..65
Robert Gerbicz, Optimal tilings for n=3..57
Gordon Hamilton, Mondrian Art Puzzles (2015).
Gordon Hamilton and Brady Haran, Mondrian Puzzle, Numberphile video (2016)
Mersenneforum.org puzzles, Mondrian art puzzles
Cooper O'Kuhn, The Mondrian Puzzle: A Connection to Number Theory, arXiv:1810.04585 [math.CO], 2018.
Cooper O'Kuhn and Todd Fellman, The Mondrian Puzzle: A Bound Concerning the M(n)=0 Case, arXiv:2006.12547 [math.NT], 2020. See also Integers (2021) Vol. 21, #A37.
Ed Pegg Jr, Mondrian Art Problem.

Cf. A050501, A278970, A279596.

Bruce Norskog corrected a(18), and a recheck by Pegg corrected a(15) and a(19). - Charles R Greathouse IV, Nov 28 2016
Correction of a(14), a(16), a(23) and new terms a(25)-a(28) from Robert Gerbicz, Nov 28 2016
a(29)-a(44) from Robert Gerbicz, Dec 02 2016
a(45)-a(47) from Robert Gerbicz added, as well as best known values to a(96).
Correction of a(45), a(46) and new terms a(48)-a(57) from Robert Gerbicz, Dec 27 2016
a(58)-a(65) from Michel Gaillard, Oct 23 2020

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 *)

cp's OEIS Frontend

A096056 Duplicate of A050501.

Views

Author

Keywords

A050499 Nearest integer to n/log(n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

JavaScript

Mathematica

PARI

A276523 Partition an n X n square into multiple non-congruent integer-sided rectangles. a(n) is the least possible difference between the largest and smallest area.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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

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