cp's OEIS Frontend

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

%I A230191 #38 Apr 19 2025 06:12:09
%S A230191 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 A230191 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 A230191 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 A230191 Decimal expansion of log( 2^(1/2)*3^(1/3)*5^(1/5) / 30^(1/30) ).
%C A230191 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 A230191 Nazardonyavi references this constant (but with a typo in the definition). - _Charles R Greathouse IV_, Nov 20 2018
%D A230191 Harold M. Edwards, Riemann's zeta function, Dover Publications, Inc., New York, 2001, pp. 281-284.
%D A230191 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 A230191 Sadegh Nazardonyavi, Improved explicit bounds for some functions of prime numbers, Functiones et Approximatio Commentarii Mathematici 58:1 (2018), pp. 7-22.
%D A230191 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 164.
%H A230191 Paolo Xausa, <a href="/A230191/b230191.txt">Table of n, a(n) for n = 0..10000</a>
%H A230191 Pafnuty Lvovich 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 A230191 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_number_theorem">Prime number theorem</a>.
%H A230191 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A230191 Equals log(6^9*10^5)/30.
%F A230191 Equals log(2)/2 + log(3)/3 + log(5)/5 - log(30)/30 = (5/6)*A230192.
%e A230191 0.921292022934090780913408449961604716417080789093030241095500286433861...
%t A230191 First[RealDigits[Log[6^9*10^5]/30, 10, 100]] (* _Paolo Xausa_, Apr 01 2024 *)
%o A230191 (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 A230191 Cf. A000040, A000720, A230192.
%K A230191 nonn,cons
%O A230191 0,1
%A A230191 _Arkadiusz Wesolowski_, Oct 11 2013
%E A230191 Better definition from _N. J. A. Sloane_, Jan 20 2019