A201542 Integer nearest f(2^n), where f(x) = Sum of ( mu(k) * H(k)/k^(3/2) * Integral Log(x^(1/k)) ) for k = 1 to infinity, where H(k) is the harmonic number sum_{i=1..k} 1/i.

2, 2, 4, 6, 11, 18, 31, 54, 96, 171, 309, 562, 1029, 1896, 3514, 6545, 12247, 23005, 43371, 82029, 155598, 295927, 564164, 1077892, 2063545, 3957761, 7603593, 14630713, 28192867, 54399529, 105097590, 203280493, 393614506, 762937782, 1480207843, 2874399615

Offset: 1

Author

Arkadiusz Wesolowski, Dec 30 2011

Keywords

• nonn

Comments

This function gives a very good approximation to the number of primes less than or equal to n.

Also note that f(2^23) - pi(2^23) = 1, f(2^31) - pi(2^31) = 25, f(2^43) - pi(2^43) = 99, f(2^58) - pi(2^58) = -53540.

Links

• Chris K. Caldwell, How Many Primes Are There?
• Eric Weisstein's World of Mathematics, Prime Counting Function
• Eric Weisstein's World of Mathematics, Prime Number Theorem

Crossrefs

Cf. A007053.

Programs

Mathematica

f[n_Integer] := Sum[N[MoebiusMu[k]*HarmonicNumber[k]/k^(3/2)*LogIntegral[n^(1/k)], 50], {k, 1, 5!}]; Table[Round[f[2^n]], {n, 36}]