Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #15 Aug 13 2023 15:24:46
%S 1,4,19,145,53,83,353,607,8171,75359,78089,79259,11657,2963,12047,
%T 378137,386197,389917,397171,2804377,11344453,11457293,11626553,
%U 11694257,11825297,11922017,12023573,12096113,12231521,12287941,6207443,6239683,3140999,9479417
%N Numerator of Sum_{k=1..n} 1/sigma(k).
%H Harvey P. Dale, <a href="/A212717/b212717.txt">Table of n, a(n) for n = 1..1000</a>
%H V. Sita Ramaiah and D. Suryanarayana, <a href="http://doi.org/10.18926/mjou/33820">Sums of reciprocals of some multiplicative functions</a>, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.
%H László Tóth, <a href="https://www.emis.de/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
%F Sum_{k=1..n} a(k)/A212718(k) ~ c * (log(n) + gamma + Sum_{p prime} (p-1)^2*beta(p)*log(p)/(p*alpha(p))) + O(log(n)^(2/3)*log(log(n))^(4/3)/n), where alpha(p) = 1 - ((p-1)^2/p) * Sum_{k>=1} 1/((p^k-1)*(p^(k+1)-1)), beta(p) = Sum_{k>=1} k/((p^k-1)*(p^(k+1)-1)), and c = Product_{p prime} alpha(p) (Sita Ramaiah and Suryanarayana, 1979). - _Amiram Eldar_, Oct 16 2022
%e 1, 4/3, 19/12, 145/84, 53/28, 83/42, 353/168, ...
%p with(numtheory): a:=n->numer(sum(1/sigma(k), k=1..n)): seq(a(n), n=1..50);
%t Numerator[Table[Sum[1/DivisorSigma[1,k],{k,1,n}],{n,1,50}]]
%t Accumulate[1/DivisorSigma[1,Range[40]]]//Numerator (* _Harvey P. Dale_, Aug 13 2023 *)
%Y Cf. A000203, A212718 (denominators).
%K nonn,frac
%O 1,2
%A _Michel Lagneau_, May 25 2012