cp's OEIS Frontend

A019529 Sum of a(n) terms of 1/sqrt(k) first strictly exceeds n.

%I A019529 #13 Jun 18 2025 07:32:10
%S A019529 1,2,3,5,7,10,14,18,22,27,33,39,45,52,60,68,76,85,95,105,115,126,138,
%T A019529 150,162,175,189,202,217,232,247,263,280,297,314,332,351,370,389,409,
%U A019529 430,451,472,494,517,540,563,587,612,637,662,688,715,741,769,797,825
%N A019529 Sum of a(n) terms of 1/sqrt(k) first strictly exceeds n.
%e A019529 Let b(k) = 1 + 1/sqrt(2) + 1/sqrt(3) + ... + 1/sqrt(k):
%e A019529 .k.......1....2.....3.....4.....5.....6.....7
%e A019529 -------------------------------------------------
%e A019529 b(k)...1.00..1.71..2.28..2.78..3.23..3.64..4.01
%e A019529 For A019529 we have:
%e A019529   n=0: smallest k is a(0) = 1 since 1.00 > 0
%e A019529   n=1: smallest k is a(1) = 2 since 1.71 > 1
%e A019529   n=2: smallest k is a(2) = 3 since 2.28 > 2
%e A019529   n=3: smallest k is a(3) = 5 since 3.23 > 3
%e A019529   n=4: smallest k is a(4) = 7 since 4.01 > 4
%e A019529 For A054040 we have:
%e A019529   n=1: smallest k is a(1) = 1 since 1.00 >= 1
%e A019529   n=2: smallest k is a(2) = 3 since 2.28 >= 2
%e A019529   n=3: smallest k is a(3) = 5 since 3.23 >= 3
%e A019529   n=4: smallest k is a(4) = 7 since 4.01 >= 4
%t A019529 s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/Sqrt[ k ], 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 75} ]
%t A019529 With[{c=N[Accumulate[Table[1/Sqrt[x],{x,1000}]]]},Table[Position[c,_?(#>n&),1,1],{n,0,1000}]]//Flatten (* _Harvey P. Dale_, Mar 19 2025 *)
%Y A019529 A054040 is another version. See also A002387, A004080.
%K A019529 nonn
%O A019529 0,2
%A A019529 _Robert G. Wilson v_
%E A019529 Edited by _N. J. A. Sloane_, Sep 01 2009