A230191 Decimal expansion of log( 2^(1/2)*3^(1/3)*5^(1/5) / 30^(1/30) ).

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

Offset

0,1

Comments

Pafnuty Lvovich Chebyshev proved in 1852 that A*x/log(x) < pi(x) < B*x/log(x) holds for all x >= x(0) with some x(0) sufficiently large, where A is the constant given above and B = 6*A/5.

Nazardonyavi references this constant (but with a typo in the definition). - Charles R Greathouse IV, Nov 20 2018

References

Harold M. Edwards, Riemann's zeta function, Dover Publications, Inc., New York, 2001, pp. 281-284.

Kolmogorov, A.N., Yushkevich, A.P. (Eds.), Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory, Probability Theory, Birkhaeser-Verlag, 1992. See p. 185. - N. J. A. Sloane, Jan 20 2019

Sadegh Nazardonyavi, Improved explicit bounds for some functions of prime numbers, Functiones et Approximatio Commentarii Mathematici 58:1 (2018), pp. 7-22.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

P. L. Chebyshev, Mémoire sur les nombres premiers, Journal de Math. Pures et Appl. 17 (1852), 366-390.

Wikipedia, Prime number theorem

Index entries for transcendental numbers

Formula

Equals log(6^9*10^5)/30.

Equals log(2)/2 + log(3)/3 + log(5)/5 - log(30)/30 = (5/6)*A230192.

Example

0.921292022934090780913408449961604716417080789093030241095500286433861...

Mathematica

First[RealDigits[Log[6^9*10^5]/30, 10, 100]] (* Paolo Xausa, Apr 01 2024 *)

Prog

(PARI) default(realprecision, 105); x=log(6^9*10^5)/3; for(n=1, 105, d=floor(x); x=(x-d)*10; print1(d, ", "));

Crossrefs

Cf. A000040, A000720, A230192.

Sequence in context: A010160 A093962 A350298 * A198984 A118322 A199507

Adjacent sequences: A230188 A230189 A230190 * A230192 A230193 A230194

Keyword

nonn,cons

Author

Arkadiusz Wesolowski, Oct 11 2013

Extensions

Better definition from N. J. A. Sloane, Jan 20 2019

Status

approved