A072980 -id:A072980 - cp's OEIS frontend

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

Ilya Gutkovskiy, May 26 2022

			1, 3/2, 11/6, 61/30, 11/5, 82/35, 171/70, 1951/770, 26133/10010, 13424/5005, 41273/15015, ...

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.

Cf. A001008, A005117, A024451, A072980, A096795, A106830, A354418 (denominators).

Cf. A001620, A059956, A306016.

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

Accumulate[1/Select[Range[43], SquareFreeQ]] // Numerator

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

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

A072983 Denominator of b(n) = Sum_{k'<=n} 1/k', where k' denotes the squarefree numbers.

Original entry on oeis.org

1, 2, 6, 6, 30, 5, 35, 35, 35, 70, 770, 770, 10010, 5005, 15015, 15015, 255255, 255255, 4849845, 4849845, 1616615, 3233230, 74364290, 74364290, 74364290, 37182145, 37182145, 37182145, 1078282205, 6469693230, 200560490130, 200560490130

Offset: 1

Benoit Cloitre, Aug 21 2002

b(n) was used by Niven in the 1970's for an alternative proof of the divergence of the sum of reciprocals of the primes.

Ivan Niven, A Proof of the Divergence of sum 1/p, The American Mathematical Monthly, Vol. 78, No. 3 (Mar., 1971), pp. 272-273.

Cf. A005117, A013928, A072980 (numerators), A354418.

Accumulate[Table[If[SquareFreeQ[n], 1, 0]/n, {n, 1, 50}]] // Denominator (* Amiram Eldar, Apr 22 2025 *)

a(n)=denominator(sum(k=1,n,issquarefree(k)/k))

a(n) = A354418(A013928(n+1)). - Amiram Eldar, Apr 22 2025

cp's OEIS Frontend

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

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