A354417 a(n) is the numerator of the sum of the reciprocals of the first n squarefree numbers.
1, 3, 11, 61, 11, 82, 171, 1951, 26133, 13424, 41273, 716656, 13871719, 4700888, 9548741, 222854273, 112857219, 3310041496, 20075905417, 628822761157, 19239404599, 9709078632, 1959180271, 73097429088, 147378388979, 445594718515, 18404305970657, 3089336006908, 133763418792581
Offset: 1
Examples
1, 3/2, 11/6, 61/30, 11/5, 82/35, 171/70, 1951/770, 26133/10010, 13424/5005, 41273/15015, ...
Links
- Robert Israel, Table of n, a(n) for n = 1..1433
- Sebastian Zuniga Alterman, Explicit averages of square-free supported functions: to the edge of the convolution method, Colloquium Mathematicum, Vol. 168 (2022), pp. 1-23; arXiv preprint, arXiv:2003.05887 [math.NT], 2020.
- Olivier Ramaré, Explicit average orders: news and problems, Banach Center Publications, Vol. 118 (2019), pp. 153-176.
Crossrefs
Programs
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Maple
s:= 0: R:= NULL: count:= 0: for x from 1 while count < 40 do if numtheory:-issqrfree(x) then s:= s + 1/x; v:= numer(s); R:= R, v; count:= count+1; fi; od: R; # Robert Israel, Mar 05 2023 -
Mathematica
Accumulate[1/Select[Range[43], SquareFreeQ]] // Numerator
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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
Formula
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