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%I #9 Aug 13 2021 18:29:23
%S 1,1,1,5,13,23,101,307,1201,1453,13463,15773,177329,203069,227093,
%T 499231,7766207,938543,16470957,17764249,56987867,60515027,1314248101,
%U 462870797,2433336849,2547743449,71763644723,74631981623,2084014033867,2161650352627
%N a(n) = numerator of Sum_{i=1..n} +-1/n, where the sign is -1 iff n is prime.
%C The denominators are given by A003418.
%C The sum diverges. In fact, the n-th partial sum is C_1 log n - C2 log log n, regardless of ordering, which can only affect c1 and c2. - _R. K. Guy_
%e The partial sums are 1, 1/2, 1/6, 5/12, 13/60, 23/60, 101/420, 307/840, 1201/2520, 1453/2520, ...
%p f:=proc(n) if isprime(n) then -1/n else 1/n fi; end; g:=proc(n) add(f(i),i=1..n); end;
%t Table[Numerator[HarmonicNumber[n] - 2*Sum[1/Prime[i], {i, 1, PrimePi[n]}]], {n, 1, 30}] (* _Stefan Steinerberger_, Feb 23 2006 *)
%t Accumulate[Table[If[PrimeQ[n],-1/n,1/n],{n,30}]]//Numerator (* _Harvey P. Dale_, Aug 13 2021 *)
%K nonn,frac
%O 1,4
%A _N. J. A. Sloane_, Feb 23 2006
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