A007514 -id:A007514 - cp's OEIS frontend

A068456 Factorial expansion of zeta(7) = Sum_{n>=1} a(n)/n!.

1, 0, 0, 0, 1, 0, 0, 0, 5, 7, 9, 5, 2, 12, 13, 10, 10, 4, 4, 4, 14, 4, 10, 14, 12, 9, 22, 9, 11, 9, 8, 14, 26, 25, 28, 22, 35, 0, 24, 0, 20, 18, 13, 21, 31, 30, 22, 24, 19, 34, 16, 42, 36, 46, 35, 46, 32, 16, 34, 53, 11, 44, 45, 49, 36, 49, 13, 53, 67, 53, 63, 11, 9, 9, 16, 37, 59, 8

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A007514.

SetDefaultRealField(RealField(250)); L:=RiemannZeta(); [Floor(Evaluate(L,7))] cat [Floor(Factorial(n)*Evaluate(L,7)) - n*Floor(Factorial((n-1))*Evaluate(L,7)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

t = Zeta[7]; s = {}; Do[n = Floor[t*i!]; t -= n/i!; AppendTo[s, n], {i, 1, 30}]; s (* Amiram Eldar, Nov 25 2018 *)
With[{b = Zeta[7]}, 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(7))\1) \\ M. F. Hasler, Nov 25 2018

default(realprecision, 250); for(n=1, 80, print1(if(n==1, floor(zeta(7)), floor(n!*zeta(7)) - n*floor((n-1)!*zeta(7))), ", ")) \\ G. C. Greubel, Nov 26 2018

def A068456(n):
    if (n==1): return floor(zeta(7))
    else: return expand(floor(factorial(n)*zeta(7)) - n*floor(factorial(n-1)*zeta(7)))
[A068456(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

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

A068457 Factorial expansion of zeta(8) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 6, 4, 3, 5, 10, 0, 1, 11, 14, 4, 2, 1, 17, 12, 19, 18, 18, 6, 7, 24, 24, 7, 9, 14, 28, 27, 14, 4, 19, 33, 24, 4, 14, 29, 21, 38, 17, 20, 5, 22, 30, 7, 13, 44, 19, 4, 19, 19, 14, 7, 48, 9, 58, 49, 17, 26, 35, 33, 36, 9, 28, 36, 54, 36, 70, 0, 33, 29, 45, 14, 46, 69

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

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

Cf. A007514, adjacent sequences.

SetDefaultRealField(RealField(250)); L:=RiemannZeta(); [Floor(Evaluate(L,8))] cat [Floor(Factorial(n)*Evaluate(L,8)) - n*Floor(Factorial((n-1))*Evaluate(L,8)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

t = Zeta[8]; s = {}; Do[n = Floor[t*i!]; t -= n/i!; AppendTo[s, n], {i, 1, 30}]; s (* Amiram Eldar, Nov 25 2018 *)
With[{b = Zeta[8]}, 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(8))\1) \\ M. F. Hasler, Nov 25 2018

default(realprecision, 250); for(n=1, 80, print1(if(n==1, floor(zeta(8)), floor(n!*zeta(8)) - n*floor((n-1)!*zeta(8))), ", ")) \\ G. C. Greubel, Nov 26 2018

def A068457(n):
    if (n==1): return floor(zeta(8))
    else: return expand(floor(factorial(n)*zeta(8)) - n*floor(factorial(n-1)*zeta(8)))
[A068457(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

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

A068458 Factorial expansion of zeta(9) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 3, 0, 8, 8, 0, 7, 4, 12, 9, 8, 11, 11, 9, 16, 15, 11, 10, 11, 1, 16, 13, 25, 24, 0, 15, 23, 12, 32, 18, 21, 20, 15, 20, 19, 18, 1, 5, 18, 20, 13, 16, 35, 6, 46, 40, 28, 9, 3, 19, 34, 14, 6, 0, 26, 48, 45, 58, 10, 0, 36, 32, 21, 30, 42, 12, 65, 54, 26, 29, 15, 46, 65

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A007514, adjacent sequences.

SetDefaultRealField(RealField(250)); L:=RiemannZeta(); [Floor(Evaluate(L,9))] cat [Floor(Factorial(n)*Evaluate(L,9)) - n*Floor(Factorial((n-1))*Evaluate(L,9)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

t = Zeta[9]; s = {}; Do[n = Floor[t*i!]; t -= n/i!; AppendTo[s, n], {i, 1, 30}]; s (* Amiram Eldar, Nov 25 2018 *)
With[{b = Zeta[9]}, 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(9))\1) \\ M. F. Hasler, Nov 25 2018

default(realprecision, 250); for(n=1, 80, print1(if(n==1, floor(zeta(9)), floor(n!*zeta(9)) - n*floor((n-1)!*zeta(9))), ", ")) \\ G. C. Greubel, Nov 26 2018

def A068458(n):
    if (n==1): return floor(zeta(9))
    else: return expand(floor(factorial(n)*zeta(9)) - n*floor(factorial(n-1)*zeta(9)))
[A068458(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

A068459 Factorial expansion of zeta(10): zeta(10) = Sum_{n>0} a(n)/n!.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 5, 0, 0, 9, 1, 3, 1, 0, 1, 11, 2, 15, 10, 18, 4, 16, 9, 20, 12, 6, 1, 23, 20, 14, 22, 0, 8, 9, 3, 26, 15, 6, 13, 11, 20, 32, 7, 12, 31, 39, 46, 36, 6, 49, 7, 10, 2, 5, 44, 11, 32, 41, 49, 21, 40, 17, 49, 62, 44, 13, 25, 67, 41, 57, 27, 13, 24, 35, 25, 43, 25, 27, 29

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..150 from Vincenzo Librandi)

Cf. A007514.

SetDefaultRealField(RealField(250)); L:=RiemannZeta(); [Floor(Evaluate(L,10))] cat [Floor(Factorial(n)*Evaluate(L,10)) - n*Floor(Factorial((n-1))*Evaluate(L,10)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

t = Zeta[10]; s = {}; Do[n = Floor[t*i!]; t -= n/i!; AppendTo[s, n], {i, 1, 30}]; s (* Amiram Eldar, Nov 25 2018 *)
With[{b = Zeta[10]}, 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(10))\1) \\ Increase realprecision (do e.g. \p500) to compute more terms. - M. F. Hasler, Nov 25 2018

default(realprecision, 500); for(n=1, 80, print1(if(n==1, floor(zeta(10)), floor(n!*zeta(10)) - n*floor((n-1)!*zeta(10))), ", ")) \\ G. C. Greubel, Nov 26 2018

def A068459(n):
    if (n==1): return floor(zeta(10))
    else: return expand(floor(factorial(n)*zeta(10)) - n*floor(factorial(n-1)*zeta(10)))
[A068459(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

Keywords cons,easy 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

A067840 Factorial expansion of e^2 : exp(2) = Sum_{n >= 0} a(n)/n!.

Original entry on oeis.org

7, 0, 0, 2, 1, 1, 4, 0, 6, 6, 6, 8, 5, 11, 5, 10, 10, 15, 16, 8, 19, 18, 15, 0, 16, 1, 2, 26, 17, 27, 23, 17, 18, 11, 24, 34, 25, 16, 27, 5, 33, 20, 11, 39, 35, 25, 39, 7, 5, 21, 27, 30, 33, 21, 34, 9, 10, 26, 32, 15, 35, 23, 6, 3, 21, 43, 50, 40, 41, 33, 1, 62, 58, 59, 12, 23, 62, 42

Offset: 0

Benoit Cloitre, Mar 10 2002

Cf. A072334, A007514.

a(0) = 7; for n>=1, a(n) = floor(n!*e^2) - n*floor((n-1)!*e^2).

Offset changed to 0 by Sean A. Irvine, Jan 09 2024

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

A068448 Factorial expansion of log(Pi) = Sum_{n>0} a(n)/n! with a(n) as large as possible.

Original entry on oeis.org

1, 0, 0, 3, 2, 2, 1, 3, 4, 5, 8, 10, 11, 7, 13, 13, 3, 14, 11, 16, 6, 9, 3, 14, 0, 16, 22, 9, 8, 26, 5, 18, 6, 3, 13, 31, 4, 27, 25, 5, 12, 1, 17, 31, 2, 4, 16, 17, 39, 15, 15, 25, 52, 40, 50, 3, 27, 32, 54, 18, 55, 10, 29, 62, 38, 4, 17, 53, 13, 24, 22, 40, 23, 11, 74, 18, 74, 31, 8

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

If a(n) is not required to be as large as possible, it isn't well defined: it can be decreased by any amount x without changing the value of the sum, if x*(n+1) is added to a(n+1), which in turn can be decreased by any arbitrary amount etc. - M. F. Hasler, Dec 04 2018

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system: Fractional values
Index to sequences related to factorial base representation of noninteger constants.

Cf. A053510 (decimal expansion).

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

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

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

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

A068448_vec(N=90,c=log(precision(Pi,N*log(N/2.4)\/2.3)))=vector(N,n,if(n>1,c=c%1*n,c)\1) \\ N*log(N/2.4)\/2.3 ~ logint(N!,10) but uses much less memory when N is big. - M. F. Hasler, Nov 28 2018

Name edited by M. F. Hasler, Dec 04 2018

A068449 Factorial expansion of log(Pi/2) = sum n>0 a(n)/n!.

Original entry on oeis.org

0, 0, 2, 2, 4, 1, 0, 7, 7, 3, 3, 6, 4, 10, 9, 1, 15, 2, 8, 10, 14, 6, 4, 7, 3, 3, 2, 2, 7, 26, 3, 30, 3, 31, 9, 29, 23, 12, 29, 3, 0, 12, 1, 11, 4, 13, 22, 17, 24, 33, 40, 34, 48, 27, 15, 5, 33, 33, 51, 48, 42, 46, 47, 38, 35, 30, 27, 1, 51, 52, 28, 25, 13, 30, 51, 14, 39, 12, 9, 58, 33

Offset: 1

Benoit Cloitre, Mar 10 2002

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

Cf. A007514, A094642 (decimal expansion).

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

Table[If[n == 1, Floor[Log[Pi/2]], Floor[n!*Log[Pi/2]] - n*Floor[(n - 1)!*Log[Pi/2]]], {n, 1, 50}] (* G. C. Greubel, Mar 21 2018 *)

for(n=1,30, print1(if(n==1, floor(log(Pi/2)), floor(n!*log(Pi/2)) - n*floor((n-1)!*log(Pi/2))), ", ")) \\ G. C. Greubel, Mar 21 2018

A068462 Factorial expansion of the cube root of 2: 2^(1/3) = Sum_{n>0} a(n)/n!.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 1, 0, 0, 1, 5, 6, 10, 0, 11, 2, 13, 9, 9, 5, 15, 3, 1, 4, 18, 16, 3, 0, 27, 6, 8, 30, 7, 25, 31, 18, 9, 18, 9, 18, 38, 2, 2, 40, 26, 30, 41, 19, 9, 45, 39, 22, 28, 44, 20, 27, 33, 7, 3, 53, 6, 5, 4, 45, 44, 32, 4, 48, 53, 50, 15, 71, 53, 53, 1, 24, 23, 48, 56, 54, 1, 16

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

Cf. A007514.

vector(30, n, if(n>1, t=t%1*n, t=2^(1/3))\1) \\ Increase realprecision (e.g., \p500) to compute more terms. - M. F. Hasler, Nov 26 2018

cp's OEIS Frontend

A068456 Factorial expansion of zeta(7) = Sum_{n>=1} a(n)/n!.

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A068457 Factorial expansion of zeta(8) = Sum_{n>=1} a(n)/n!.

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A068458 Factorial expansion of zeta(9) = Sum_{n>=1} a(n)/n!.

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A068459 Factorial expansion of zeta(10): zeta(10) = Sum_{n>0} a(n)/n!.

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

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A067840 Factorial expansion of e^2 : exp(2) = Sum_{n >= 0} a(n)/n!.

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

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