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A354417 a(n) is the numerator of the sum of the reciprocals of the first n squarefree numbers.

%I A354417 #17 Mar 06 2023 01:54:47
%S A354417 1,3,11,61,11,82,171,1951,26133,13424,41273,716656,13871719,4700888,
%T A354417 9548741,222854273,112857219,3310041496,20075905417,628822761157,
%U A354417 19239404599,9709078632,1959180271,73097429088,147378388979,445594718515,18404305970657,3089336006908,133763418792581
%N A354417 a(n) is the numerator of the sum of the reciprocals of the first n squarefree numbers.
%H A354417 Robert Israel, <a href="/A354417/b354417.txt">Table of n, a(n) for n = 1..1433</a>
%H A354417 Sebastian Zuniga Alterman, <a href="https://doi.org/10.4064/cm8337-11-2020">Explicit averages of square-free supported functions: to the edge of the convolution method</a>, Colloquium Mathematicum, Vol. 168 (2022), pp. 1-23; <a href="https://arxiv.org/abs/2003.05887">arXiv preprint</a>, arXiv:2003.05887 [math.NT], 2020.
%H A354417 Olivier Ramaré, <a href="https://hal.archives-ouvertes.fr/hal-02585977/">Explicit average orders: news and problems</a>, Banach Center Publications, Vol. 118 (2019), pp. 153-176.
%F A354417 a(n)/A354418(n) ~ (6/Pi^2) * (log(n) + c) + O*(1.044/sqrt(n)), where f = O*(g) means |f| <= g and c = gamma + 2 * Sum_{p prime} log(p)/(p^2-1) = A001620 + 2 * A306016 = 1.71713765109059847340... (Ramaré, 2019; Alterman, 2022). - _Amiram Eldar_, Oct 29 2022
%e A354417 1, 3/2, 11/6, 61/30, 11/5, 82/35, 171/70, 1951/770, 26133/10010, 13424/5005, 41273/15015, ...
%p A354417 s:= 0: R:= NULL: count:= 0:
%p A354417 for x from 1 while count < 40 do
%p A354417   if numtheory:-issqrfree(x) then
%p A354417     s:= s + 1/x;
%p A354417     v:= numer(s);
%p A354417     R:= R, v;
%p A354417     count:= count+1;
%p A354417   fi;
%p A354417 od:
%p A354417 R; # _Robert Israel_, Mar 05 2023
%t A354417 Accumulate[1/Select[Range[43], SquareFreeQ]] // Numerator
%o A354417 (PARI) a(n) = my(i=0, s=0); for(x=1, oo, if(core(x)==x, s+=1/x; i++; if(i==n, return(numerator(s))))) \\ _Felix Fröhlich_, May 26 2022
%Y A354417 Cf. A001008, A005117, A024451, A072980, A096795, A106830, A354418 (denominators).
%Y A354417 Cf. A001620, A059956, A306016.
%K A354417 nonn,frac
%O A354417 1,2
%A A354417 _Ilya Gutkovskiy_, May 26 2022