[seqfan] Re: relation of A019529 and A054040, partial sums of 1/sqrt(j)

[Rainer Rosenthal]

David Wilson wrote:
> Yes. Looks like A054040 is in error. Should be deaded and merged into 
> A019529.

Right so.

A019529(n) = 1, 2, 3, 5, 7, ...
A054040(n) = 1, 3, 5, 7, ...

Let b(k) = 1 + 1/sqrt(2) + 1(sqrt(3) + ... + 1/sqrt(k)
and
a(n) = smallest k such that b(k) > n, i = 0,1,2,3,...

Sequence a(n) is what A054040 and A019529 are meant to be,
according to their definition.

 k       1    2     3     4     5     6     7
-------------------------------------------------
b(k)   1.00  1.71  2.28  2.78  3.23  3.64  4.01

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

Thus sequence a(n) starts with 1, 2, 3, 5, 7, which is what
A054040 fails to do.

Cheers,
Rainer