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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.
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%I #17 Apr 03 2023 10:36:13

%S 4,25,168,1226,9585,78521,664652,5761512,50847348,455050385,

%T 4118051652,37607908133,346065524108,3204941711340,29844570436484,

%U 279238341185832,2623557156537070,24739954282695698,234057667295619287,2220819602542218793

%N 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 The sequence gives exactly the values of pi(10^n) for n = 1 to 3.

%C A228724 gives the difference between A006880 and this sequence.

%H David Baugh, <a href="/A226945/b226945.txt">Table of n, a(n) for n = 1..100</a>

%H Chris K. Caldwell, <a href="https://t5k.org/howmany.shtml">How Many Primes Are There?</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeNumberTheorem.html">Prime Number Theorem</a>

%t 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 Cf. A006880, A057834, A226744, A057754, A190802, A057793, A201542, A228724.

%K nonn

%O 1,1

%A _Arkadiusz Wesolowski_, Aug 31 2013