%I
%S 0,2,5,9,14,18,24,32,37,45,56,65,70,80,90,102,122,130,136,146,160,171,
%T 192,205,228,256,260,268,279,292,308,324,343,372,391,411,444,480,513,
%U 518,528,537,550,569,584,605,640,649,672,705,744,773,792,823,858,904
%N a(n) is the smallest k such that Sum_{i=0..k} 1/A001316(i) >= n.
%C Note that A001316(n) = Sum_{k=0..n} mod(C(n,k),2).
%C Therefore, the infinite series Sum_{n>=0} 1/A001316(n) is a special case of Sum_{n>=0} 1/Sum_{k=0..n} ((mod(C(n,k),2)*x^k) for x=1 (cf. A001317, A100307, A100308, etc.). For x>1 this series is convergent, while for x=1 it is divergent. It would be interesting to have asymptotic estimates for a(n).
%t a[n_] := Module[{k=0, s=1}, While[s < n, k++; s += (1/Numerator[2^k / k!])] ; k]; Array[a, 60] (* _Amiram Eldar_, Dec 04 2018 *)
%o (PARI) f(n) = numerator(2^n / n!); \\ A001316
%o a(n) = my(k=0, s=1/f(0)); while (s < n, k++; s += 1/f(k)); k; \\ _Michel Marcus_, Dec 04 2018
%Y Cf. A001316, A000120.
%K nonn
%O 1,2
%A _Vladimir Shevelev_, Jan 06 2014
%E More terms from _Peter J. C. Moses_, Jan 06 2014
