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

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Examples

			1.945910149055313305105352743443179729637084729581861188459390149937579...
		

References

• 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.

Links

• 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

Crossrefs

Cf. A016735 Continued fraction.

Programs

Mathematica

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

PARI

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

Formula

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