A336112 a(n) is the least number k such that the Sum_{i=0..k} sqrt(k) equals or exceeds n.

0, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23

Offset: 0

Author

Robert G. Wilson v, Jul 08 2020

Keywords

• nonn

Comments

Inspired by A045880.

Let c = (9/4)^(1/3) = (3/2)^(2/3) ~ 1.310370697..., then a(n) ~ c*n^(2/3).

a(10^k) for k>= 0: 1, 6, 28, 131, 608, 2823, 13104, 60822, 282311, 1310371, 6082202, 28231081, 131037070, 608220200, ..., .

Examples

			a(0) = 0 since the sqrt(0) = 0;
a(1) = 1 since the sqrt(0) + sqrt(1) = 1;
a(2) = 2 since the sqrt(0) + sqrt(1) + sqrt(2) ~ 2.41421... which exceeds 2;
a(3) = 3 since the sqrt(0) + sqrt(1) + sqrt(2) + sqrt(3) ~ 4.146264... which easily exceeds 3;
a(4) = 3 because the sqrt(0) + sqrt(1) + sqrt(2) + sqrt(3) ~ 4.146264... which barely exceeds 4; etc.
		

Crossrefs

Cf. A025224, A045880, A073333.

Programs

Mathematica

f[n_] := Block[{k = s = 0}, While[s < n, k++; s = s + Sqrt@k]; k]; Array[f, 75, 0]

PARI

a(n) = my(s=0, k=0); while ((s+=sqrt(k)) < n, k++); k; \\ Michel Marcus, Jul 09 2020

Formula

a(k*n) ~ k^(2/3)*a(n).