A016733 -id:A016733 - cp's OEIS frontend

A016628 Decimal expansion of log(5).

1, 6, 0, 9, 4, 3, 7, 9, 1, 2, 4, 3, 4, 1, 0, 0, 3, 7, 4, 6, 0, 0, 7, 5, 9, 3, 3, 3, 2, 2, 6, 1, 8, 7, 6, 3, 9, 5, 2, 5, 6, 0, 1, 3, 5, 4, 2, 6, 8, 5, 1, 7, 7, 2, 1, 9, 1, 2, 6, 4, 7, 8, 9, 1, 4, 7, 4, 1, 7, 8, 9, 8, 7, 7, 0, 7, 6, 5, 7, 7, 6, 4, 6, 3, 0, 1, 3, 3, 8, 7, 8, 0, 9, 3, 1, 7, 9, 6, 1

Offset: 1

N. J. A. Sloane

			1.60943791243410037460075933322618763952560135426851772191264789... - _Harry J. Smith_, May 16 2009

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
Horace S. Uhler, Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17. Proc. Nat. Acad. Sci. U. S. A. 26, (1940). 205-212.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Simon Plouffe, The natural logarithm of 5 to 10000 digits
Index entries for transcendental numbers

Cf. A016733 (continued fraction). - Harry J. Smith, May 16 2009

RealDigits[Log[5], 10, 125][[1]] (* Alonso del Arte, Oct 04 2014 *)

default(realprecision, 20080); x=log(5); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016628.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009

From Peter Bala, Nov 11 2019: (Start)
log(5) = 2*sqrt(2)*Integral_{t = 0..sqrt(2)/2} (1 - t^2)/(1 + t^4) dt.
log(5) = Sum_{n >= 0} (4*n+5)/((4*n+1)*(4*n+3))*(-1/4)^n.
log(5) = (1/4)*Sum_{n >= 0} ( 8/(8*n+1) - 4/(8*n+3) - 2/(8*n+5) + 1/(8*n+7) )*(1/16)^n, a BBP-type formula. (End)
log(5) = 2*Sum_{n >= 0} (-1)^(n*(n+1)/2)*1/((2*n+1)*2^n). - Peter Bala, Oct 29 2020
log(5) = Integral_{x = 0..1} (x^4 - 1)/log(x) dx. - Peter Bala, Nov 14 2020
log(5) = 2*Sum_{n >= 1} 1/(n*P(n, 3/2)*P(n-1, 3/2)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(5) = 1.6094379124341003(29...), correct to 16 decimal places. - Peter Bala, Mar 18 2024

A322333 Factorial expansion of log(5) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 1, 0, 2, 3, 0, 5, 4, 4, 8, 3, 3, 2, 0, 7, 8, 0, 7, 11, 1, 18, 16, 3, 10, 16, 21, 17, 13, 20, 12, 16, 8, 27, 24, 28, 12, 9, 34, 21, 3, 9, 8, 41, 42, 35, 31, 4, 4, 37, 38, 9, 20, 10, 31, 24, 34, 44, 21, 16, 19, 24, 4, 22, 22, 47, 8, 28, 26, 32, 22, 28, 56, 44, 16, 61, 38, 3, 25, 52, 35, 73, 55, 8, 42, 25, 21, 62, 61, 7, 89, 5, 74, 89, 57, 33, 60, 13, 75, 95, 66

Offset: 1

G. C. Greubel, Dec 03 2018

nonn

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

Cf. A016628 (decimal expansion), A016733 (continued fraction).

Cf. A067882 (log(2)), A322334 (log(3)), A068460 (log(7)), A068461 (log(11)).

SetDefaultRealField(RealField(250));  [Floor(Log(5))] cat [Floor(Factorial(n)*Log(5)) - n*Floor(Factorial((n-1))*Log(5)) : n in [2..80]];

With[{b = Log[5]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = log(5); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

def a(n):
    if (n==1): return floor(log(5))
    else: return expand(floor(factorial(n)*log(5)) - n*floor(factorial(n-1)*log(5)))
[a(n) for n in (1..80)]

A076594 First occurrence of n as a term in the continued fraction for log(5).

Original entry on oeis.org

1, 20, 6, 11, 19, 12, 24, 26, 32, 66, 112, 45, 68, 318, 64, 52, 58, 41, 101, 62, 1168, 299, 291, 189, 74, 110, 122, 200, 287, 755, 734, 73, 1619, 71, 415, 268, 191, 1700, 27, 17, 547, 468, 3224, 3510, 690, 78, 3064, 258, 487, 1801, 2138, 3911, 155, 811, 1121

Offset: 1

Benoit Cloitre, Oct 20 2002

Cf. A016733, A032523.

Flatten[Table[Position[ContinuedFraction[Log[5],6000],n,{1},1],{n,60}]] (* Harvey P. Dale, Jun 26 2013 *)

default(realprecision, 1500); v=contfrac(log(5)); a(n)=if(n<0,0,s=1; while(abs(n-component(v,s))>0,s++); s)

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A016628 Decimal expansion of log(5).

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A322333 Factorial expansion of log(5) = Sum_{n>=1} a(n)/n!.

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A076594 First occurrence of n as a term in the continued fraction for log(5).

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