A007514 -id:A007514 - cp's OEIS frontend

A004605 Expansion of Pi in base 6.

3, 0, 5, 0, 3, 3, 0, 0, 5, 1, 4, 1, 5, 1, 2, 4, 1, 0, 5, 2, 3, 4, 4, 1, 4, 0, 5, 3, 1, 2, 5, 3, 2, 1, 1, 0, 2, 3, 0, 1, 2, 1, 4, 4, 4, 2, 0, 0, 4, 1, 1, 5, 2, 5, 2, 5, 5, 3, 3, 1, 4, 2, 0, 3, 3, 3, 1, 3, 1, 1, 3, 5, 5, 3, 5, 1, 3, 1, 2, 3, 3, 4, 5, 5, 3, 3, 4, 1, 0, 0, 1, 5, 1, 5, 4, 3, 4, 4, 4, 0, 1, 2, 3, 4, 3

Offset: 1

N. J. A. Sloane

			3.05033005141512410523441405312532110230...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), this sequence (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).

Cf. A007514.

RealDigits[Pi, 6, 105][[1]]
Table[ResourceFunction["NthDigit"][Pi, n, 6], {n, 1, 105}] (* Joan Ludevid ,Aug 17 2022;easy to compute a(10000000)=0 with this function;requires Mathematica 12.0+ *)

A004604 Expansion of Pi in base 5.

Original entry on oeis.org

3, 0, 3, 2, 3, 2, 2, 1, 4, 3, 0, 3, 3, 4, 3, 2, 4, 1, 1, 2, 4, 1, 2, 2, 4, 0, 4, 1, 4, 0, 2, 3, 1, 4, 2, 1, 1, 1, 4, 3, 0, 2, 0, 3, 1, 0, 0, 2, 2, 0, 0, 3, 4, 4, 4, 1, 3, 2, 2, 1, 1, 0, 1, 0, 4, 0, 3, 3, 2, 1, 3, 4, 4, 0, 0, 4, 3, 2, 4, 4, 4, 0, 1, 4, 4, 1, 0, 4, 2, 3, 3, 4, 1, 3, 3, 0, 1, 1, 3, 2

Offset: 1

N. J. A. Sloane

			3.03232214303343241124122404140231421114...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wadim Zudilin, A BBP-style computation for π in base 5, arXiv:2409.10097 [math.NT], 2024.

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), this sequence (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).

Cf. A007514.

RealDigits[Pi, 5, 100][[1]]
Table[ResourceFunction["NthDigit"][Pi, n, 5], {n, 1, 100}] (* Joan Ludevid, Aug 17 2022;easy to compute a(10000000)=0 with this function;requires Mathematica 12.0+ *)

A004606 Expansion of Pi in base 7.

Original entry on oeis.org

3, 0, 6, 6, 3, 6, 5, 1, 4, 3, 2, 0, 3, 6, 1, 3, 4, 1, 1, 0, 2, 6, 3, 4, 0, 2, 2, 4, 4, 6, 5, 2, 2, 2, 6, 6, 4, 3, 5, 2, 0, 6, 5, 0, 2, 4, 0, 1, 5, 5, 4, 4, 3, 2, 1, 5, 4, 2, 6, 4, 3, 1, 0, 2, 5, 1, 6, 1, 1, 5, 4, 5, 6, 5, 2, 2, 0, 0, 0, 2, 6, 2, 2, 4, 3, 6, 1, 0, 3, 3, 0, 1, 4, 4, 3, 2, 3, 3, 6, 3, 1, 0, 1, 1, 3

Offset: 1

N. J. A. Sloane

			3.06636514320361341102634022446522266435...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), this sequence (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).

Cf. A007514.

RealDigits[Pi, 7, 105][[1]]
Table[ResourceFunction["NthDigit"][Pi, n, 7], {n, 1, 105}] (* Joan Ludevid, Sep 13 2022; easy to compute a(10000000)=5 with this function;requires Mathematica 12.0+ *)

A006941 Expansion of Pi in base 8.

Original entry on oeis.org

3, 1, 1, 0, 3, 7, 5, 5, 2, 4, 2, 1, 0, 2, 6, 4, 3, 0, 2, 1, 5, 1, 4, 2, 3, 0, 6, 3, 0, 5, 0, 5, 6, 0, 0, 6, 7, 0, 1, 6, 3, 2, 1, 1, 2, 2, 0, 1, 1, 1, 6, 0, 2, 1, 0, 5, 1, 4, 7, 6, 3, 0, 7, 2, 0, 0, 2, 0, 2, 7, 3, 7, 2, 4, 6, 1, 6, 6, 1, 1, 6, 3, 3, 1, 0, 4, 5, 0, 5, 1, 2, 0, 2, 0, 7, 4, 6, 1, 6, 1, 5, 0, 0, 2, 3

Offset: 1

N. J. A. Sloane

			3.1103755242102643021514230630505600670...

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 614.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to the number Pi

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), this sequence (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).

Cf. A007514.

convert(evalf(Pi), octal, 120);  # Alois P. Heinz, Dec 16 2018

RealDigits[ N[ Pi, 105], 8] [[1]]
Table[ResourceFunction["NthDigit"][Pi, n, 8], {n, 1, 105}] (* Joan Ludevid, Sep 13 2022; easy to compute a(10000000)=1 with this function; requires Mathematica 12.0+ *)

a(n) = 4*A004601(3n) + 2*A004601(3n+1) + 1*A004601(3n+2). - Jason Kimberley, Nov 06 2012

More terms from Michel ten Voorde, Apr 14 2001

A068438 Expansion of Pi in base 13.

Original entry on oeis.org

3, 1, 10, 12, 1, 0, 4, 9, 0, 5, 2, 10, 2, 12, 7, 7, 3, 6, 9, 12, 0, 11, 11, 8, 9, 12, 12, 9, 8, 8, 3, 2, 7, 8, 2, 9, 8, 3, 5, 8, 11, 3, 7, 0, 1, 6, 0, 3, 0, 6, 1, 3, 3, 12, 10, 5, 10, 12, 11, 10, 5, 7, 6, 1, 4, 11, 6, 5, 11, 4, 1, 0, 0, 2, 0, 12, 2, 2, 11, 4, 12, 7, 1, 4, 5, 7, 10, 9, 5, 5, 10, 5

Offset: 1

Benoit Cloitre, Mar 09 2002

			3.1ac1049052a2c77369c0aa89cc988327829835...

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), this sequence (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).

Cf. A007514.

RealDigits[Pi, 13, 111][[1]] (* slightly modified by Robert G. Wilson v, Dec 13 2017 *)
Table[ResourceFunction["NthDigit"][Pi, n, 13], {n, 1, 111}] (* Joan Ludevid, Oct 11 2022; easy to compute a(10000000)=1 with this function; requires Mathematica 12.0+ *)

A075874 Pi = Sum_{n >= 1} a(n)/n!, with largest possible a(n).

Original entry on oeis.org

3, 0, 0, 3, 1, 5, 6, 5, 0, 1, 4, 7, 8, 0, 6, 7, 10, 7, 10, 4, 10, 6, 16, 1, 11, 20, 3, 18, 12, 9, 13, 18, 21, 14, 34, 27, 11, 27, 33, 36, 18, 5, 18, 5, 23, 39, 1, 10, 42, 28, 17, 20, 51, 8, 42, 47, 0, 27, 23, 16, 52, 32, 52, 53, 24, 43, 61, 64, 18, 17, 11, 0, 53, 14, 62

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Nov 02 2001 and Oct 20 2002

nonn

What is meant is the expansion in the factorial number system, cf. links. The formula itself is not sufficient to define the terms uniquely: a(n) can be decreased by any amount x if x*(n+1) is added to a(n+1). - M. F. Hasler, Nov 26 2018

			Pi = 3/1! + 0/2! + 0/3! + 3/4! + 1/5! + ...

Hans Havermann, Table of n, a(n) for n = 1..10000
D. E. Knuth, The Art of Computer Programming, Vol.2, 3rd ed., Addison-Wesley, 2014, ISBN 978-0321635761, p.209.
Eric Weisstein's World of Mathematics, Harmonic Expansion
Wikipedia, Factorial number system

Essentially same as A007514.
Pi in base n: A004601 to A004608, A000796, A068436 to A068440, A062964.
Cf. A068452-A068464.

SetDefaultRealField(RealField(250)); R:=RealField(); [Floor(Pi(R))] cat [Floor(Factorial(n)*Pi(R)) - n*Floor(Factorial((n-1))*Pi(R)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

Digits := 120; M := proc(a,n) local i,b,c; b := a; c := [ floor(b) ]; for i from 1 to n-1 do b := b-c[ i ]/i!; c := [ op(c), floor(b*(i+1)!) ]; od; c; end: t1 := M(Pi,100); A075874 := n->t1[n+1];

p = N[Pi, 1000]; Do[k = Floor[p*n! ]; p = p - k/n!; Print[k], {n, 1, 75}]
With[{b = Pi}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

x=Pi;vector(floor((y->y/log(y))(default(realprecision))),n,t=n!;k=floor(x*t);x-=k/t;k) \\ Charles R Greathouse IV, Jul 15 2011

vector(30,n,if(n>1,t=t%1*n,t=Pi)\1) \\ Increase realprecision (e.g., \p500) to compute more terms. - M. F. Hasler, Nov 25 2018

default(realprecision, 250); b = Pi; for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 26 2018

def A075874(n):
    if (n==1): return floor(pi)
    else: return expand(floor(factorial(n)*pi) - n*floor(factorial(n-1)*pi))
[A075874(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

a(1)=3; for n >= 2, a(n) = floor(n!*Pi) - n*floor((n-1)!*Pi). - Benoit Cloitre, Mar 10 2002

A068461 Factorial, or factoradic, expansion of log(11) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible.

Original entry on oeis.org

2, 0, 2, 1, 2, 4, 3, 3, 1, 2, 4, 0, 3, 13, 1, 12, 12, 13, 1, 16, 16, 0, 16, 12, 10, 9, 1, 23, 3, 22, 0, 28, 11, 14, 23, 16, 0, 14, 6, 1, 1, 14, 4, 25, 43, 0, 29, 10, 41, 19, 47, 14, 0, 51, 10, 47, 37, 45, 46, 56, 57, 45, 10, 32, 61, 15, 9, 67, 5, 9, 22, 25, 65, 56, 24, 12, 71, 9, 57

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

			log(11) = 2 + 0/2! + 2/3! + 1/4! + 2/5! + 4/6! + 3/7! + 3/8! + 1/9! + ...

Cf. A016634 (decimal expansion), A016739 (continued fraction).
Cf. A007514 vs. A075874 for factoradic expansion.
Cf. A067882 (log(2)), A322334 (log(3)), A322333 (log(5)), A068460 (log(7)).

SetDefaultRealField(RealField(250));  [Log(11)] cat [Floor(Factorial(n)*Log(11)) - n*Floor(Factorial((n-1))*Log(11)) : n in [2..80]]; // G. C. Greubel, Dec 05 2018

With[{b = Log[11]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 05 2018 *)

vector(30, n, if(n>1, t=t%1*n, t=log(11))\1) \\ Increase realprecision (e.g., \p500) to compute more terms. - M. F. Hasler, Nov 25 2018

default(realprecision, 250); b = log(11); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Dec 05 2018

def a(n):
    if n==1: return floor(log(11))
    else: return expand(floor(factorial(n)*log(11)) - n*floor(factorial(n-1)*log(11)))
[a(n) for n in (1..80)] # G. C. Greubel, Dec 05 2018

Name edited and keyword cons,easy removed by M. F. Hasler, Nov 26 2018

A068464 Factorial expansion of Gamma(1/4) = Sum_{n>=1} a(n)/n! with largest possible a(n), and Gamma = Euler's gamma function.

Original entry on oeis.org

3, 1, 0, 3, 0, 0, 3, 0, 5, 3, 2, 7, 0, 2, 8, 9, 16, 3, 1, 15, 18, 8, 20, 7, 23, 8, 10, 11, 28, 29, 24, 30, 3, 16, 10, 8, 31, 11, 30, 35, 5, 5, 38, 32, 31, 42, 13, 17, 43, 3, 41, 27, 1, 14, 26, 52, 38, 22, 55, 46, 6, 35, 46, 34, 24, 52, 52, 64, 62, 25, 46, 56, 3, 71, 70, 22, 53, 63, 53

Offset: 1

Benoit Cloitre, Mar 10 2002

			Gamma(1/4) = A068466 = 3.6256099... = 3/1! + 1/2! + 0 + 3/4! + 0 + 0 + 3/7! + 0 + 5/9! + 3/10! + 2/11! + ... - _M. F. Hasler_, Nov 26 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for factorial base representation

Cf. A007514, A068466 (decimal expansion), A068463.

SetDefaultRealField(RealField(250));  [Floor(Gamma(1/4))] cat [Floor(Factorial(n)*Gamma(1/4)) - n*Floor(Factorial((n-1))*Gamma(1/4)) : n in [2..80]]; // G. C. Greubel, Nov 27 2018

r:= Gamma[1/4]; Table[If[n == 1, Floor[r], Floor[n!*r]- n*Floor[(n-1)!*r] ], {n,1,100}] (* G. C. Greubel, Mar 29 2018 *)

default(realprecision, 250); b = gamma(1/4); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Mar 29 2018

A068464(N=90,c=gamma(precision(.25,logint(N!,10)+1)))=vector(N,n,if(n>1,c=c%1*n,c)\1) \\ - M. F. Hasler, Nov 26 2018

def A068464(n):
    if (n==1): return floor(gamma(1/4))
    else: return expand(floor(factorial(n)*gamma(1/4)) - n*floor(factorial(n-1)*gamma(1/4)))
[A068464(n) for n in (1..80)] # G. C. Greubel, Nov 27 2018

a(n) = floor(n!*Gamma(1/4)) - n*floor((n-1)!*Gamma(1/4)), for n > 1. - M. F. Hasler, Nov 26 2018

A068451 Factorial expansion of the golden ratio (1+sqrt(5))/2 = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 1, 0, 2, 4, 0, 6, 7, 1, 1, 8, 1, 6, 0, 11, 0, 10, 5, 6, 9, 15, 20, 10, 15, 1, 18, 5, 13, 9, 0, 13, 15, 2, 27, 28, 2, 32, 36, 11, 4, 34, 37, 0, 4, 32, 10, 4, 4, 32, 46, 39, 37, 2, 20, 27, 8, 54, 27, 45, 9, 26, 18, 59, 0, 22, 63, 41, 54, 65, 61, 45, 51, 61, 31, 68, 48, 34, 39, 71, 59

Offset: 1

Benoit Cloitre, Mar 10 2002

Cf. A001622 (decimal expansion).

Cf. A075874 and A007514.

SetDefaultRealField(RealField(250));  [Floor((1+Sqrt(5))/2)] cat [Floor(Factorial(n)*(1+Sqrt(5))/2) - n*Floor(Factorial((n-1))*(1+Sqrt(5))/2) : n in [2..80]]; // G. C. Greubel, Mar 21 2018

With[{b = GoldenRatio}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Mar 21 2018 *)

default(realprecision, 250); b = (1+sqrt(5))/2; for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Mar 21 2018

A068451(N=90,c=precision(sqrt(5)+1,logint(N!,10))/2)=vector(N,n,if(n>1,c=c%1*n,c)\1) \\ M. F. Hasler, Nov 27 2018

def A068451(n):
    if (n==1): return floor(golden_ratio)
    else: return expand(floor(factorial(n)*golden_ratio) - n*floor(factorial(n-1)*golden_ratio))
[A068451(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

A068455 Factorial expansion of zeta(6) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 3, 3, 2, 4, 5, 6, 5, 9, 14, 11, 3, 4, 0, 15, 5, 7, 10, 17, 11, 14, 12, 22, 4, 17, 21, 15, 26, 21, 9, 3, 23, 0, 4, 31, 39, 21, 13, 26, 16, 25, 27, 13, 27, 21, 19, 46, 17, 21, 25, 50, 21, 44, 55, 23, 20, 22, 10, 49, 37, 5, 55, 51, 39, 40, 63, 2, 6, 17, 61, 52, 9, 21

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A007514.

SetDefaultRealField(RealField(250)); L:=RiemannZeta(); [Floor(Evaluate(L,6))] cat [Floor(Factorial(n)*Evaluate(L,6)) - n*Floor(Factorial((n-1))*Evaluate(L,6)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

t = Zeta[6]; s = {}; Do[n = Floor[t*i!]; t -= n/i!; AppendTo[s, n], {i, 1, 30}]; s (* Amiram Eldar, Nov 25 2018 *)
With[{b = Zeta[6]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

vector(30,n,if(n>1,t=t%1*n,t=zeta(6))\1) \\ M. F. Hasler, Nov 25 2018

default(realprecision, 250); for(n=1, 80, print1(if(n==1, floor(zeta(6)), floor(n!*zeta(6)) - n*floor((n-1)!*zeta(6))), ", ")) \\ G. C. Greubel, Nov 26 2018

def A068455(n):
    if (n==1): return floor(zeta(6))
    else: return expand(floor(factorial(n)*zeta(6)) - n*floor(factorial(n-1)*zeta(6)))
[A068455(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

Name edited and keywords cons and easy removed by M. F. Hasler, Nov 25 2018

cp's OEIS Frontend

A004605 Expansion of Pi in base 6.

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A004604 Expansion of Pi in base 5.

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A004606 Expansion of Pi in base 7.

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A006941 Expansion of Pi in base 8.

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A068438 Expansion of Pi in base 13.

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A075874 Pi = Sum_{n >= 1} a(n)/n!, with largest possible a(n).

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A068461 Factorial, or factoradic, expansion of log(11) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible.

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