A059562 -id:A059562 - cp's OEIS frontend

A059561 Beatty sequence for log(Pi).

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81

Offset: 1

Mitch Harris, Jan 22 2001

a(n) is the largest integer m such that e^m < Pi^n. - Stanislav Sykora, May 29 2015

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 A059562.
Cf. A000796 (Pi), A001113 (e), A053510 (log(Pi)).
Cf. A022932 (characteristic function).

Floor[Range[100]*Log[Pi]] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=log(Pi); for (n = 1, 2000, write("b059561.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = A004777(n+1), 1 <= n < 83. - R. J. Mathar, Oct 05 2008

a(n) = floor(n*log(Pi)). - Michel Marcus, Jan 04 2015

A022932 a(n) is the number of powers Pi^m between e^n and e^(n+1).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

nonn

Characteristic function of A059561. - Antti Karttunen, Sep 22 2017

Hans Havermann & Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for characteristic functions

Cf. A059562 (positions of zeros after the initial a(0)=0), A059561 (of ones).

t = Table[IntegerPart[i/Log[Pi]] - IntegerPart[(i - 1)/Log[Pi]], {i, 1000000}]; (* Hans Havermann, Sep 22 2017 *)

a(n)=(n+1)\log(Pi) - n\log(Pi) \\ Charles R Greathouse IV, Jan 16 2017

More terms from Antti Karttunen, Sep 22 2017

A249329 First row of spectral array W(log(Pi)).

Original entry on oeis.org

1, 7, 8, 55, 62, 435, 497, 3440, 3937, 27208, 31145, 215199, 246344, 1702099, 1948443, 13462620

Offset: 1

Colin Barker, Dec 03 2014

log(Pi) = 1.144729885849400174143427351353058711647294812915311571513623...

The sequence is generated from the Beatty sequence (A059561) and from the complement of the Beatty sequence (A059562) for log(Pi).

A. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discrete Mathematics 126 (1994) 137-149.

Cf. A059561, A059562.

\\ Row i of the generalized Wythoff array W(h),
\\   where h is an irrational number between 1 and 2,
\\   and m is the number of terms in the vectors b and c.
row(h, i, m) = {
  if(h<=1 || h>=2, print("Invalid value for h"); return);
  my(
    b=vector(m, n, floor(n*h)),       \\ Beatty sequence for h
    c=vector(m, n, floor(n*h/(h-1))), \\ Complement of b
    w=[b[b[i]], c[b[i]]],
    j=3
  );
  while(1,
    if(j%2==1,
      if(w[j-1]<=#b, w=concat(w, b[w[j-1]]), return(w))
    ,
      if(w[j-2]<=#c, w=concat(w, c[w[j-2]]), return(w))
    );
    j++
  )
}
allocatemem(10^9)
default(realprecision, 100)
row(log(Pi), 1, 10^7)

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A059561 Beatty sequence for log(Pi).

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A022932 a(n) is the number of powers Pi^m between e^n and e^(n+1).

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A249329 First row of spectral array W(log(Pi)).

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PARI