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%I #19 Dec 21 2015 14:10:33
%S 0,0,1,1,3,3,5,5,5,5,7,7,10,10,10,10,13,13,16,16,16,16,19,19,19,19,19,
%T 19,22,22,26,26,26,26,26,26,29,29,29,29,33,33,37,37,37,37,40,40,40,40,
%U 40,40,44,44,44,44,44,44,49,49,53,53,53,53,53,53,57,57,57,57,61,61,65,65,65,65
%N Floor of the Chebyshev function theta(n): a(n) = floor(Sum_{primes p <= n } log(p)).
%C The old entry with this sequence number was a duplicate of A002325.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, see Chap. 22.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.35. (For inequalities, etc.)
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>, Ill. Journ. Math. 6 (1962) 64-94.
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="/A000720/a000720.html">Approximate formulas for some functions of prime numbers</a> (scan of some key pages from an ancient annotated photocopy)
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0457373-7">Sharper bounds for the Chebyshev functions theta(x) and psi(x)</a>, Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 243-269.
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://dx.doi.org/10.1090/S0025-5718-1976-0457374-X">Sharper bounds for the Chebyshev functions theta (x) and psi (x). II</a>. Math. Comp. 30 (1976), number 134, 337-360.
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://dx.doi.org/10.1090/S0025-5718-76-99659-9">Corrigendum: "Sharper bounds for the Chebyshev functions theta (x) and psi (x). II" (Math. Comput. 30 (1976), number 134, 337-360)</a>, Math. Comp. 30 (1976), number 136, 900.
%F a(n) ~ n by the prime number theorem. - _Charles R Greathouse IV_, Aug 02 2012
%p (Maple for A035158, A057872, A083535:)
%p Digits:=2000;
%p tf:=[]; tr:=[]; tc:=[];
%p for n from 1 to 120 do
%p t2:=0;
%p j:=pi(n);
%p for i from 1 to j do t2:=t2+log(ithprime(i)); od;
%p tf:=[op(tf),floor(evalf(t2))];
%p tr:=[op(tr),round(evalf(t2))];
%p tc:=[op(tc),ceil(evalf(t2))];
%p od:
%Y Cf. A057872, A083535, A016040 (records), A000040 (places of records)
%K nonn
%O 1,5
%A _N. J. A. Sloane_, Oct 02 2008
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