Harmonic number sum-优快云博客

本文通过VBScript实现两个关于调和级数的问题：一是找出使得调和级数H(k)≥n的最小k值；二是计算从n到k的调和级数之和也大于等于n时的最小k值。提供了具体的代码实现及运行结果。

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Problem1: Least k such that H(k) >= n, where H(k) is the harmonic number sum_{i=1..k} 1/i.

Sub Harmonic_number_sum() Dim s As Double, i As Long, n As Long n = 1 For i = 1 To 300000000 s = s + 1 / i If s >= n Then Debug.Print "a(" & n & ")=" & i: n = n + 1 Next End Sub

It returns:

a(1)=1
a(2)=4
a(3)=11
a(4)=31
a(5)=83
a(6)=227
a(7)=616
a(8)=1674
a(9)=4550
a(10)=12367
a(11)=33617
a(12)=91380
a(13)=248397
a(14)=675214
a(15)=1835421
a(16)=4989191
a(17)=13562027
a(18)=36865412
a(19)=100210581
a(20)=272400600

Problem2: Least k such that H(k) >= n, where H(k) is the harmonic number sum_{i=n..k} 1/i.

Note: The sequence was published at :http://www.research.att.com/~njas/sequences/A168214

Sub Harmonic_number_sum() Dim s As Double, i As Long, j As Long, n As Long For n = 1 To 15 s = 0 For i = n To 1000000000 s = s + 1 / i If s >= n Then Exit For Next Debug.Print "a(" & n & ")=" & i: Next End Sub

It returns:

a(1)=1
a(2)=11
a(3)=51
a(4)=192
a(5)=669
a(6)=2222
a(7)=7135
a(8)=22374
a(9)=68916
a(10)=209348
a(11)=628916
a(12)=1872269
a(13)=5531641
a(14)=16238866
a(15)=47410139