cp's OEIS Frontend

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

%I A336112 #20 Aug 27 2021 02:18:21
%S A336112 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,
%T A336112 12,13,13,13,14,14,14,14,15,15,15,15,16,16,16,16,17,17,17,17,18,18,18,
%U A336112 18,19,19,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22,22,23,23,23,23
%N A336112 a(n) is the least number k such that the Sum_{i=0..k} sqrt(k) equals or exceeds n.
%C A336112 Inspired by A045880.
%C A336112 Let c = (9/4)^(1/3) = (3/2)^(2/3) ~ 1.310370697..., then a(n) ~ c*n^(2/3).
%C A336112 a(10^k) for k>= 0: 1, 6, 28, 131, 608, 2823, 13104, 60822, 282311, 1310371, 6082202, 28231081, 131037070, 608220200, ..., .
%F A336112  a(k*n) ~ k^(2/3)*a(n).
%e A336112 a(0) = 0 since the sqrt(0) = 0;
%e A336112 a(1) = 1 since the sqrt(0) + sqrt(1) = 1;
%e A336112 a(2) = 2 since the sqrt(0) + sqrt(1) + sqrt(2) ~ 2.41421... which exceeds 2;
%e A336112 a(3) = 3 since the sqrt(0) + sqrt(1) + sqrt(2) + sqrt(3) ~ 4.146264... which easily exceeds 3;
%e A336112 a(4) = 3 because the sqrt(0) + sqrt(1) + sqrt(2) + sqrt(3) ~ 4.146264... which barely exceeds 4; etc.
%t A336112 f[n_] := Block[{k = s = 0}, While[s < n, k++; s = s + Sqrt@k]; k]; Array[f, 75, 0]
%o A336112 (PARI) a(n) = my(s=0, k=0); while ((s+=sqrt(k)) < n, k++); k; \\ _Michel Marcus_, Jul 09 2020
%Y A336112 Cf. A025224, A045880, A073333.
%K A336112 nonn
%O A336112 0,3
%A A336112 _Robert G. Wilson v_, Jul 08 2020