A333342 Least k such that f(k) < 0, where f(1) = n and f(k) = f(k-1)*e^(1/k) - 1 for k > 1.

2, 3, 7, 14, 27, 53, 103, 199, 383, 738, 1422, 2739, 5274, 10156, 19556, 37657, 72511, 139625, 268857, 517702, 996870, 1919539, 3696200, 7117277, 13704787, 26389471, 50814668, 97846995, 188410842, 362797502, 698590519, 1345182118, 2590236887

Offset

0,1

Comments

If f(k) = f(k-1)*c^(1/k) - 1 for k > 1 and c > e = 2.718281828..., then as long as f(1) is large enough, f(k) can always be positive.

It appears that lim_{n->oo} a(n+1)/a(n) = 1.92556595....

Links

Table of n, a(n) for n=0..32.

Formula

a(n) <= A002387(n+1). Proof: since e < (1 + 1/(k-1))^k for any k > 1, f(k)/k < f(k-1)/(k-1) - 1/k < f(1) - Sum_{i=2..k} 1/i. If k = A002387(n+1), then Sum_{i=2..k} 1/i > n and f(k)/k < 0.

Example

For n=0, f(1) = 0 and f(2) = -1 < 0. So, a(0) = 2.
For n=1, f(1) = 1, f(2) = sqrt(e)-1 and f(3) = -0.094636534... < 0. So, a(1) = 3.

Prog

(PARI) a(n) = {my(k=1, f=n); while(f>0, f=f*exp(1/k++)-1); k+!n; }

Crossrefs

Cf. A001113 (e), A002387.

Sequence in context: A054194 A138651 A131300 * A078043 A294627 A293326

Adjacent sequences: A333339 A333340 A333341 * A333343 A333344 A333345

Keyword

nonn,more

Author

Jinyuan Wang, Mar 26 2020

Status

approved