A075874 -id:A075874 - cp's OEIS frontend

A083303 Let Pi= sum(k>=0, u(k)/k!) where u(k)>=0 are integer (u(k)=A075874(k)), then sequence gives values of m such that u(m)=0.

Original entry on oeis.org

1, 2, 3, 9, 14, 57, 72, 76, 553, 1489, 6839, 44620

Offset: 1

Benoit Cloitre, Jun 03 2003

Integers n such that frac((n-1)!*Pi) < 1/n.

A075874(a(n))=0

a(10)-a(12) from Max Alekseyev, Jun 17 2011

A007514 Pi = Sum_{n >= 0} a(n)/n!.

Original entry on oeis.org

3, 0, 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: 0

N. J. A. Sloane, Robert G. Wilson v

nonn

The current name does not define a(n) without ambiguity. It is meant that for each n, a(n) is the largest integer such that the remainder of Pi - (partial sum up to n) remains positive. This leads to the FORMULA given below. - M. F. Hasler, Mar 20 2017

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hans Havermann, Table of n, a(n) for n = 0..10000
Index entries for sequences related to the number Pi

Essentially same as A075874.

Pi in base n: A004601 to A004608, A000796, A068436 to A068440, A062964.

p = N[Pi, 1000]; Do[k = Floor[p*n! ]; p = p - k/n!; Print[k], {n, 0, 75} ]

x=Pi;vector(floor((y->y/log(y))(default(realprecision))),n,t=(n-1)!;k=floor(x*t);x-=k/t;k) \\ Charles R Greathouse IV, Jul 15 2011

C=1/Pi;x=0;vector(primepi(default(realprecision)),n,-x*n--+x=n!\C) \\ M. F. Hasler, Mar 20 2017

a(n) = floor(n!*Pi) - n*floor((n-1)!*Pi) for all n > 0. - M. F. Hasler, Mar 20 2017

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

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

A068463 Factorial expansion of Gamma(3/4) = Sum_{n>=1} a(n)/n! where Gamma is Euler's gamma function.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 2, 0, 7, 2, 1, 5, 1, 12, 12, 12, 12, 5, 7, 9, 4, 20, 10, 9, 6, 17, 20, 18, 7, 6, 11, 9, 24, 3, 22, 8, 24, 33, 35, 33, 31, 12, 0, 27, 6, 31, 37, 37, 27, 31, 6, 24, 7, 17, 12, 1, 39, 41, 51, 48, 21, 8, 15, 26, 46, 52, 43, 39, 7, 21, 60, 24, 58, 21, 57, 58, 36, 60, 25, 7

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

			Gamma(3/4) = 1 + 0/2! + 1/3! + 1/4! + 2/5! + 0/6! + 2/7! + ...

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

Cf. A075874, A068465 (decimal expansion), A068464 (Gamma(1/4)).

SetDefaultRealField(RealField(250)); [Floor(Gamma(3/4))] cat [Floor(Factorial(n)*Gamma(3/4)) - n*Floor(Factorial((n-1))*Gamma(3/4)) : n in [2..80]]; // G. C. Greubel, Nov 27 2018

With[{b = Gamma[3/4]}, Table[If[n == 1, Floor[b], Floor[n!*b]-n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 27 2018 *)

A068463(N=90,c=gamma(precision(.75,logint(N!,10)+1)))=vector(N,n,if(n>1,c=c%1*n,c)\1) \\ - M. F. Hasler, Nov 26 2018

default(realprecision, 250); b = gamma(3/4); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 27 2018

def A068463(n):
    if (n==1): return floor(gamma(3/4))
    else: return expand(floor(factorial(n)*gamma(3/4)) - n*floor(factorial(n-1)*gamma(3/4)))
[A068463(n) for n in (1..80)] # G. C. Greubel, Nov 27 2018

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

A068450 Factorial expansion of sqrt(Pi) = Sum_{n>0} a(n)/n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 1, 1, 3, 0, 5, 10, 6, 8, 12, 0, 10, 0, 12, 9, 6, 12, 22, 21, 24, 3, 14, 21, 13, 24, 21, 11, 8, 22, 27, 3, 8, 1, 36, 21, 27, 15, 2, 41, 22, 34, 8, 0, 4, 8, 39, 48, 27, 38, 22, 0, 23, 49, 19, 27, 29, 28, 40, 33, 21, 62, 7, 67, 33, 8, 30, 18, 60, 73, 61, 72, 42, 59, 22

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

			sqrt(Pi) = 1 + 1/2! + 1/3! + 2/4! + 2/5! + 4/6! + 1/7! + ...

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

Cf. A075874, A002161 (decimal expansion).

SetDefaultRealField(RealField(250));  R:= RealField(); [Floor(Sqrt(Pi(R)))] cat [Floor(Factorial(n)*Sqrt(Pi(R))) - n*Floor(Factorial((n-1))*Sqrt(Pi(R))) : n in [2..30]]; // G. C. Greubel, Mar 21 2018

Table[If[n == 1, Floor[Sqrt[Pi]], Floor[n!*Sqrt[Pi]] - n*Floor[(n - 1)!*Sqrt[Pi]]], {n, 1, 50}] (* G. C. Greubel, Mar 21 2018 *)

default(realprecision, 250); for(n=1,30, print1(if(n==1, floor(sqrt(Pi)), floor(n!*sqrt(Pi)) - n*floor((n-1)!*sqrt(Pi))), ", ")) \\ G. C. Greubel, Mar 21 2018

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

def A068450(n):
    if (n==1): return floor(sqrt(pi))
    else: return expand(floor(factorial(n)*sqrt(pi)) - n*floor(factorial(n-1)*sqrt(pi)))
[A068450(n) for n in (1..80)] # G. C. Greubel, Nov 27 2018

Keyword cons removed by R. J. Mathar, Jul 23 2009

A068453 Factorial expansion of sqrt(e) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 1, 0, 3, 2, 5, 0, 4, 3, 9, 8, 2, 8, 0, 10, 15, 2, 10, 8, 19, 12, 4, 18, 23, 8, 4, 21, 15, 17, 1, 11, 19, 7, 25, 15, 3, 20, 5, 24, 25, 35, 9, 12, 25, 26, 22, 23, 11, 43, 46, 6, 0, 25, 27, 30, 6, 14, 20, 33, 5, 30, 23, 42, 4, 11, 19, 55, 63, 43, 12, 52, 51, 22, 29, 11, 8, 19, 35, 25

Offset: 1

Benoit Cloitre, Mar 10 2002

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

Cf. A067840 (e^2), A075874 (Pi).

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

With[{b = Sqrt[E]}, 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=exp(.5))\1) \\ M. F. Hasler, Nov 25 2018

default(realprecision, 250); b = sqrt(exp(1)); 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 A068453(n):
    if (n==1): return floor(sqrt(e))
    else: return expand(floor(factorial(n)*sqrt(e)) - n*floor(factorial(n-1)*sqrt(e)))
[A068453(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

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

A068454 Factorial expansion of zeta(5) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible.

Original entry on oeis.org

1, 0, 0, 0, 4, 2, 4, 0, 8, 3, 4, 9, 10, 5, 3, 12, 4, 1, 10, 0, 6, 19, 0, 19, 10, 21, 19, 16, 3, 27, 24, 12, 12, 14, 7, 33, 27, 15, 28, 15, 7, 15, 7, 21, 13, 29, 16, 44, 39, 27, 39, 17, 6, 18, 2, 21, 21, 35, 29, 12, 13, 6, 39, 14, 1, 23, 55, 34, 10, 42, 70, 14, 42, 26, 74, 64, 12, 42, 14

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000 (terms 1..300 from Vincenzo Librandi)
Wikipedia, Factorial number system
Index entries for factorial base representation of noninteger constants

Cf. A075874 (same for Pi), A007514 (different variant).

Cf. A067279 (zeta(2)), A067277 (zeta(3)), A068447 (zeta(4)), A068455 (zeta(6)), A068456 (zeta(7)), A068457 (zeta(8)), A068458 (zeta(9)), A068459 (zeta(10)).

SetDefaultRealField(RealField(250)); b:=Evaluate(RiemannZeta(),5); [n eq 1 select Floor(b) else Floor(Factorial(n)*b) - n*Floor(Factorial(n)*b/n) : n in [1..100]]; // G. C. Greubel, Nov 26 2018

t = Zeta[5]; s = {}; Do[n = Floor[t*i!]; t -= n/i!; AppendTo[s, n], {i, 1, 30}]; s (* Amiram Eldar, Nov 25 2018 *)
With[{b = Zeta[5]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

vector(N=100, n, if(n>1, c=c%1*n, c=zeta(precision(5.,N*log(N/2.7)\2.3+3)))\1) \\ Specific a(n) can be computed via the FORMULA. For repeated use the value of c can be stored as a global variable, to be re-computed with higher precision when log_10(n!) exceeds its precision. - M. F. Hasler, Nov 25 2018

b=zeta(5)
@cached_function
def A068454(n):
    if n == 1: return floor(b)
    else: return expand(floor(factorial(n)*b) - n*floor(factorial(n-1)*b))
[A068454(n) for n in (1..100)] # G. C. Greubel, Nov 26 2018

a(n) = floor(c*n!) - n*floor(c*(n-1)!) = floor(frac(c*(n-1)!)*n) for n > 1, with c = zeta(5). - M. F. Hasler, Dec 20 2018

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

A240455 Primorial expansion of Pi.

Original entry on oeis.org

3, 0, 0, 4, 1, 8, 1, 0, 8, 19, 13, 10, 28, 29, 23, 30, 9, 32, 4, 26, 12, 27, 75, 28, 45, 30, 47, 65, 91, 83, 9, 92, 123, 44, 73, 32, 140, 102, 28, 75, 108, 30, 139, 4, 127, 88, 57, 182, 207, 172, 80, 126, 150, 232, 227, 19, 256, 238, 195, 44, 56, 58, 131, 160, 243, 222, 22, 47, 30, 226, 312, 130, 161, 68, 358, 52, 250, 152, 15, 38, 120, 195, 120, 263, 412, 115, 412, 427, 284, 361, 121, 413, 355, 75, 473, 355, 10, 177, 101, 71

Offset: 0

Albert Lau, Apr 05 2014

The primorial expansion a(n) of a real number x is defined as x = Sum_{i>=0} a(i) / prime(i)# where a(0) = floor(x) and 0 <= a(i) < prime(i) for all i > 0.

			Pi = 3/prime(0)# + 0/prime(1)# + 0/prime(2)# + 4/prime(3)# + 1/prime(4)# + 8/prime(5)# + ... where prime(n)# = A002110(n) is the n-th primorial number.

Cf. A000796 (decimal expansion), A075874 (factorial number system expansion).

pe = Block[{x = #, $MaxExtraPrecision = \[Infinity]},
       Do[x = Prime[i] (x - Sow[x // Floor]) // Expand, {i, #2 - 1}];
       x // Floor // Sow] // Reap // Last // Last // Function;
pe[\[Pi], 100]

x(0) = Pi; a(n) = floor(x(n)) where x(n + 1) = prime(n + 1) * (x(n) - a(n)) and prime(n) = A000040(n) is the n-th prime number. [corrected by Rémy Sigrist, Jan 06 2019]

A068452 Pi^2 = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

9, 1, 2, 0, 4, 2, 0, 6, 4, 0, 4, 11, 6, 4, 14, 8, 12, 6, 18, 12, 12, 14, 13, 2, 7, 20, 12, 2, 16, 21, 25, 26, 29, 19, 7, 3, 20, 3, 38, 7, 12, 19, 37, 1, 23, 32, 19, 32, 38, 45, 45, 27, 44, 34, 14, 49, 35, 29, 30, 57, 57, 18, 56, 48, 33, 19, 44, 35, 12, 56, 28, 38, 64, 35, 10, 45, 35, 0

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

For the fractional part, this corresponds to the factoradic (or factorial base, or harmonic) expansion, but the integer part 9 = 3! + 2! + 1! would be [1, 1, 1] in factorial base, cf. A007623(9) = 111. - M. F. Hasler, Nov 27 2018

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
Eric Weisstein's World of Mathematics, Harmonic Expansion.
Wikipedia, Factorial number system: Fractional values
Index to sequences related to factorial base representation of noninteger constants.

Cf. A002388 (decimal expansion of Pi^2).

Similar expansions: A068450 (sqrt(Pi)), A075874 (Pi), A007514 (different variant for Pi).

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

fexp := proc(x) local xres,a,n ; xres := x ; a := [] ; for n from 1 to 100 do a := [op(a),floor(n!*xres)] ; xres := xres-op(-1,a)/n! ; od: a ; end: Digits := 400 ; fexp(evalf(Pi^2)) ; Digits := 600 ; fexp(evalf(Pi^2)) ; # R. J. Mathar, Sep 30 2008

p=N[Pi, 10000]^2; Do[k=Floor[p n!]; p=p - k / n!; Print[k], {n, 1000}] (* Vincenzo Librandi, Nov 24 2018 *)
With[{b = Pi^2}, Table[If[n == 1, Floor[b], Floor[n!*b] -n*Floor[(n- 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

default(realprecision, 250); b = Pi^2; for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 26 2018

A068452(N=90, c=precision(Pi^2,logint(N!,10)))=vector(N, n, if(n>1, c=c%1*n, c)\1) \\ M. F. Hasler, Nov 27 2018

def A068452(n):
    if (n==1): return floor(pi^2)
    else: return expand(floor(factorial(n)*pi^2) - n*floor(factorial(n-1)*pi^2))
[A068452(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

Corrected beginning at 3rd term by R. J. Mathar, Sep 30 2008

cp's OEIS Frontend

A083303 Let Pi= sum(k>=0, u(k)/k!) where u(k)>=0 are integer (u(k)=A075874(k)), then sequence gives values of m such that u(m)=0.

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A007514 Pi = Sum_{n >= 0} a(n)/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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A068451 Factorial expansion of the golden ratio (1+sqrt(5))/2 = Sum_{n>=1} a(n)/n!.

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A068463 Factorial expansion of Gamma(3/4) = Sum_{n>=1} a(n)/n! where Gamma is Euler's gamma function.

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A068450 Factorial expansion of sqrt(Pi) = Sum_{n>0} a(n)/n!.

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A068453 Factorial expansion of sqrt(e) = Sum_{n>=1} a(n)/n!.

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