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

1,5

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

Table of n, a(n) for n=1..76.

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.

Formula

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

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:

Crossrefs

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

Sequence in context: A129972 A130829 A196386 * A196172 A123313 A131507

Adjacent sequences: A035155 A035156 A035157 * A035159 A035160 A035161

Keyword

nonn

Author

N. J. A. Sloane, Oct 02 2008

Status

approved