A364415 a(n) is the least k such that Sum_{j=1..k} 1/(j^(1 + 1/j)) >= n.

0, 1, 6, 22, 65, 185, 512, 1402, 3825, 10412, 28318, 76995, 209314, 568995, 1546713, 4204428, 11428848, 31066858, 84448506, 229554871, 623994868, 1696193945, 4610733216, 12533272358, 34068966559

Offset

0,3

Comments

As Sum_{j=1..k} 1/(j^(1 + 1/j)) diverges this sequence is well-defined.

References

R. P. Boas, Growth of partial sums of divergent series, Mathematics of Computation, 31 (1977), 257-264.

Links

Table of n, a(n) for n=0..24.

Michael Penn, A notorious Calculus problem, YouTube video, 2023.

Formula

a(n) >= A004080(n).

Conjecture: Limit_{n->oo} a(n)/a(n-1) = e. - Hugo Pfoertner, Jul 29 2023

Example

Let f(m) = Sum_{j=1..m} 1/(j^(1 + 1/j)) and n = 2. Then 1.906406... f(5) < n = 2 <= f(6) = 2.030045839... . So 6 is the smallest m such that f(m) >= 2. Therefore a(2) = 6.

Mathematica

a={0}; sum=0; k=1; For[n=1, n<=8, n++, While[ sum<=n, If[(sum+=1/(k^(1+1/k)))>=n, AppendTo[a, k]]; k++]]; a (* Stefano Spezia, Jul 24 2023 *)

Prog

(PARI) a(n) = if(n == 0, return(0)); my(t = 0); for(i = 1, oo, t+= 1/(i^(1+1/i)); if(t >= n, return(i)))

Crossrefs

Cf. A002387, A004080.

Sequence in context: A099855 A347435 A003469 * A189418 A027992 A271389

Adjacent sequences: A364412 A364413 A364414 * A364416 A364417 A364418

Keyword

nonn,more

Author

David A. Corneth, Jul 23 2023

Extensions

a(13)-a(18) from Hugo Pfoertner, Jul 29 2023

a(19)-a(24) from Sela Fried, Jul 03 2024

Status

approved