A059561 -id:A059561 - cp's OEIS frontend

A004777 Numbers not congruent to 7 mod 8.

0, 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

Offset: 1

N. J. A. Sloane

Numbers that are congruent to {0, 1, 2, 3, 4, 5, 6} mod 8.
Numbers n such that binary expansion does not end 111.
Complement of A004771. - Michel Marcus, Sep 11 2015

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A004771, A059561, A132270.

[n-1+Floor((n-1)/7) : n in [1..100]]; // Wesley Ivan Hurt, Sep 11 2015

[n: n in [0..100] | not n mod 8 in [7]]; // Vincenzo Librandi, Sep 11 2015

A004777:=n->n-1+floor((n-1)/7): seq(A004777(n), n=1..100); # Wesley Ivan Hurt, Sep 11 2015

DeleteCases[Range[0,100],?(Mod[#,8]==7&)] (* _Harvey P. Dale, Apr 01 2011 *)
Select[Range[0, 100], ! MemberQ[{7}, Mod[#, 8]] &] (* Vincenzo Librandi, Sep 12 2015 *)

A004777=n->(n-1)*8\7 \\ M. F. Hasler, Nov 02 2013

G.f.: x^2*(1+x+x^2+x^3+x^4+x^5+2*x^6) / ((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
a(n) = floor((n-1)*8/7). - M. F. Hasler, Nov 02 2013
From Wesley Ivan Hurt, Sep 11 2015: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n>8.
a(n) = n - 1 + A132270(n). (End)
From Wesley Ivan Hurt, Jul 20 2016: (Start)
a(n) = (56*n - 77 + (n mod 7) + ((n+1) mod 7) + ((n+2) mod 7) + ((n+3) mod 7) + ((n+4) mod 7) + ((n+5) mod 7) - 6*((n+6) mod 7))/49.
a(7k) = 8k-2, a(7k-1) = 8k-3, a(7k-2) = 8k-4, a(7k-3) = 8k-5, a(7k-4) = 8k-6, a(7k-5) = 8k-7, a(7k-6) = 8k-8. (End)

Edited by N. J. A. Sloane Aug 31 2009 at the suggestion of R. J. Mathar.

A059562 Beatty sequence for log(Pi)/(log(Pi)-1).

Original entry on oeis.org

7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 181, 189, 197, 205, 213, 221, 229, 237, 245, 253, 261, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 355, 363, 371, 379, 387, 395, 403, 411, 419, 427

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 A059561.

Cf. A053510.

Floor[Range[100]*(1 + 1/(Log[Pi] - 1))] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=log(Pi)/(log(Pi) - 1); for (n = 1, 2000, write("b059562.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

A059562(n,c=1-1/log(Pi))=n\c \\ Use \pXX to set sufficiently large precision. - M. F. Hasler, Oct 06 2014

a(n) = floor(n*(1 + 1/(A053510 - 1))). - Paolo Xausa, Jul 05 2024

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

A022931 Number of e^m between Pi^n and Pi^(n+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Clark Kimberling

			Pi^5 = 306.01968478528145326274131... and Pi^6 = 961.389193575304437...; in between them we find e^6 = 403.4287934927351226... and no other powers of e with integer exponents. Hence a(5) = 1.
Pi^6 = 961.389193575304437... and Pi^7 = 3020.2932277767920675142...; in between them we find e^7 = 1096.63315842845859926372... and e^8 = 2980.957987041728274743592... Hence a(6) = 2.

Cf. A000796 (Pi), A001113 (e), A053510 (log(Pi)), A059561 (floor(n*log(Pi))).

Digits:= 30:
log_Pi:= evalf(log(Pi));
a:= n-> floor((n+1)*log_Pi) -floor(n*log_Pi):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 21 2018

Table[Floor[(n + 1)Log[Pi]] - Floor[n Log[Pi]], {n, 0, 99}] (* Alonso del Arte, Dec 21 2018 *)

val logPi = Math.log(Math.PI); for (n <- 0 to 99) yield (Math.floor(logPi  * (n + 1)) - Math.floor(logPi * n)).toInt // Alonso del Arte, Dec 21 2018

a(n) = floor((n + 1) log Pi) - floor(n log Pi). - Alonso del Arte, Dec 20 2018

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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A004777 Numbers not congruent to 7 mod 8.

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A059562 Beatty sequence for log(Pi)/(log(Pi)-1).

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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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A022931 Number of e^m between Pi^n and Pi^(n+1).

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

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