A035158 Floor of the Chebyshev function theta(n): a(n) = floor(Sum_{primes p <= n } log(p)).

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, 19, 22, 22, 26, 26, 26, 26, 26, 26, 29, 29, 29, 29, 33, 33, 37, 37, 37, 37, 40, 40, 40, 40, 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

Offset: 1

Author

N. J. A. Sloane, Oct 02 2008

Keywords

• nonn

Comments

The old entry with this sequence number was a duplicate of A002325.

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, see Chap. 22.
• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.35. (For inequalities, etc.)

Links

• J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. Journ. Math. 6 (1962) 64-94.
• J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers (scan of some key pages from an ancient annotated photocopy)
• J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x), Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 243-269.
• J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta (x) and psi (x). II. Math. Comp. 30 (1976), number 134, 337-360.
• J. Barkley Rosser and Lowell Schoenfeld, Corrigendum: "Sharper bounds for the Chebyshev functions theta (x) and psi (x). II" (Math. Comput. 30 (1976), number 134, 337-360), Math. Comp. 30 (1976), number 136, 900.

Crossrefs

Cf. A057872, A083535, A016040 (records), A000040 (places of records)

Programs

Maple

(Maple for A035158, A057872, A083535:)
Digits:=2000;
tf:=[]; tr:=[]; tc:=[];
for n from 1 to 120 do
t2:=0;
j:=pi(n);
for i from 1 to j do t2:=t2+log(ithprime(i)); od;
tf:=[op(tf),floor(evalf(t2))];
tr:=[op(tr),round(evalf(t2))];
tc:=[op(tc),ceil(evalf(t2))];
od:

Formula

a(n) ~ n by the prime number theorem. - Charles R Greathouse IV, Aug 02 2012