A230191 - OEIS

%I #33 Apr 01 2024 17:13:18

%S 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,

%T 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,

%U 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

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

%C 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.

%C Nazardonyavi references this constant (but with a typo in the definition). - _Charles R Greathouse IV_, Nov 20 2018

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

%D 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

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

%H Paolo Xausa, <a href="/A230191/b230191.txt">Table of n, a(n) for n = 0..10000</a>

%H P. L. Chebyshev, <a href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A19_0.pdf">Mémoire sur les nombres premiers</a>, Journal de Math. Pures et Appl. 17 (1852), 366-390.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_number_theorem">Prime number theorem</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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

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

%e 0.921292022934090780913408449961604716417080789093030241095500286433861...

%t First[RealDigits[Log[6^9*10^5]/30, 10, 100]] (* _Paolo Xausa_, Apr 01 2024 *)

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

%Y Cf. A000040, A000720, A230192.

%K nonn,cons

%O 0,1

%A _Arkadiusz Wesolowski_, Oct 11 2013

%E Better definition from _N. J. A. Sloane_, Jan 20 2019