A016735 -id:A016735 - cp's OEIS frontend

A016630 Decimal expansion of log(7).

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

Offset: 1

N. J. A. Sloane

			1.945910149055313305105352743443179729637084729581861188459390149937579...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
Uhler, Horace S.; 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].
Index entries for transcendental numbers

Cf. A016735 Continued fraction.

First[RealDigits[Log[7], 10, 100]] (* Paolo Xausa, Mar 21 2024 *)

default(realprecision, 20080); x=log(7); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016630.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009

From Peter Bala, Nov 11 2019: (Start)
log(7) = 2*sqrt(3)*Integral_{t = 0..sqrt(3)/3} (1 - t^4)/(1 + t^6) dt.
log(7) = (8/9)*Sum_{n >= 0} (12*n+11)/((6*n+1)*(6*n+5))*(-1/27)^n.
log(7) = 6*Sum_{n >= 0} ( 243/(12*n+1) - 27/(12*n+5) - 9/(12*n+7) + 1/(12*n+11) )*(1/729)^(n+1), a BPP-type formula. (End)
log(7) = 2*Sum_{n >= 1} 1/(n*P(n, 4/3)*P(n-1, 4/3)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(7) = 1.945910149055(27...), correct to 12 decimal places. - Peter Bala, Mar 18 2024

A068460 Factorial expansion of log(7) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 0, 3, 0, 8, 8, 2, 11, 1, 5, 11, 1, 7, 1, 11, 16, 12, 12, 13, 5, 4, 26, 19, 12, 20, 0, 18, 14, 22, 6, 29, 0, 27, 16, 23, 23, 23, 34, 27, 4, 27, 18, 0, 10, 27, 42, 24, 9, 16, 6, 52, 2, 38, 44, 30, 51, 61, 7, 19, 0, 45, 18, 51, 43, 54, 7, 15, 69, 44, 51, 9, 74, 5, 69

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

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

Cf. A016630 (decimal expansion), A016735 (continued fraction).

Cf. A067882 (log(2)), A322334 (log(3)), A322333 (log(5)), A068461 (log(11)).

SetDefaultRealField(RealField(250));  [Floor(Log(7))] cat [Floor(Factorial(n)*Log(7)) - n*Floor(Factorial((n-1))*Log(7)) : n in [2..80]]; // G. C. Greubel, Dec 05 2018

Digits:=200: a:=n->`if`(n=1,floor(log(7)),floor(factorial(n)*log(7))-n*floor(factorial(n-1)*log(7))); seq[120](a(n),n=1..80); # Muniru A Asiru, Dec 06 2018

With[{b = Log[7]}, 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(7))\1) \\ Increase realprecision (e.g., \p500) to compute more terms. - M. F. Hasler, Nov 25 2018

default(realprecision, 250); b = log(7); 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(7))
    else: return expand(floor(factorial(n)*log(7)) - n*floor(factorial(n-1)*log(7)))
[a(n) for n in (1..80)] # G. C. Greubel, Dec 05 2018

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

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A016630 Decimal expansion of log(7).

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A068460 Factorial expansion of log(7) = Sum_{n>=1} a(n)/n!.

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