A235333 a(n) is the smallest k such that Sum_{i=0..k} 1/A001316(i) >= n.

0, 2, 5, 9, 14, 18, 24, 32, 37, 45, 56, 65, 70, 80, 90, 102, 122, 130, 136, 146, 160, 171, 192, 205, 228, 256, 260, 268, 279, 292, 308, 324, 343, 372, 391, 411, 444, 480, 513, 518, 528, 537, 550, 569, 584, 605, 640, 649, 672, 705, 744, 773, 792, 823, 858, 904

Offset: 1

Author

Vladimir Shevelev, Jan 06 2014

Keywords

• nonn

Comments

Note that A001316(n) = Sum_{k=0..n} mod(C(n,k),2).

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).

Crossrefs

Cf. A001316, A000120.

Programs

Mathematica

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 *)

PARI

f(n) = numerator(2^n / n!); \\ A001316
a(n) = my(k=0, s=1/f(0)); while (s < n, k++; s += 1/f(k)); k; \\ Michel Marcus, Dec 04 2018

Extensions

More terms from Peter J. C. Moses, Jan 06 2014