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

1, 2, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 202, 217, 232, 247, 263, 280, 297, 314, 332, 351, 370, 389, 409, 430, 451, 472, 494, 517, 540, 563, 587, 612, 637, 662, 688, 715, 741, 769, 797, 825

Offset: 0

Author

Robert G. Wilson v

Keywords

• nonn

Examples

			Let b(k) = 1 + 1/sqrt(2) + 1/sqrt(3) + ... + 1/sqrt(k):
.k.......1....2.....3.....4.....5.....6.....7
-------------------------------------------------
b(k)...1.00..1.71..2.28..2.78..3.23..3.64..4.01
For A019529 we have:
  n=0: smallest k is a(0) = 1 since 1.00 > 0
  n=1: smallest k is a(1) = 2 since 1.71 > 1
  n=2: smallest k is a(2) = 3 since 2.28 > 2
  n=3: smallest k is a(3) = 5 since 3.23 > 3
  n=4: smallest k is a(4) = 7 since 4.01 > 4
For A054040 we have:
  n=1: smallest k is a(1) = 1 since 1.00 >= 1
  n=2: smallest k is a(2) = 3 since 2.28 >= 2
  n=3: smallest k is a(3) = 5 since 3.23 >= 3
  n=4: smallest k is a(4) = 7 since 4.01 >= 4
		

Crossrefs

A054040 is another version. See also A002387, A004080.

Programs

Mathematica

s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/Sqrt[ k ], 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 75} ]
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 *)

Extensions

Edited by N. J. A. Sloane, Sep 01 2009