cp's OEIS Frontend

A226945 Integer nearest f(10^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.

%I A226945 #18 Feb 16 2025 08:33:20
%S A226945 4,25,168,1226,9585,78521,664652,5761512,50847348,455050385,
%T A226945 4118051652,37607908133,346065524108,3204941711340,29844570436484,
%U A226945 279238341185832,2623557156537070,24739954282695698,234057667295619287,2220819602542218793
%N A226945 Integer nearest f(10^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.
%C A226945 The sequence gives exactly the values of pi(10^n) for n = 1 to 3.
%C A226945 A228724 gives the difference between A006880 and this sequence.
%H A226945 David Baugh, <a href="/A226945/b226945.txt">Table of n, a(n) for n = 1..100</a>
%H A226945 Chris K. Caldwell, <a href="https://t5k.org/howmany.shtml">How Many Primes Are There?</a>
%H A226945 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>
%H A226945 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeNumberTheorem.html">Prime Number Theorem</a>
%t A226945 f[n_Integer] := Sum[N[MoebiusMu[k]*HarmonicNumber[k]/k^(3/2)*LogIntegral[n^(1/k)], 50], {k, 5!}]; Table[Round[f[10^n]], {n, 20}]
%Y A226945 Cf. A006880, A057834, A226744, A057754, A190802, A057793, A201542, A228724.
%K A226945 nonn
%O A226945 1,1
%A A226945 _Arkadiusz Wesolowski_, Aug 31 2013