

A054040


a(n) terms of series {1/sqrt(j)} are >= n.


6



1, 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
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OFFSET

1,2


COMMENTS

In many cases the first differences have the form {2k, 2k, 2k, 2k+1} (A004524). In such cases the second differences are {0, 0, 1, 1}. See A082915 for the exceptions. In as many as these, the first differences have the form {2k1, 2k1, 2k1, 2k}.  Robert G. Wilson v, Apr 18 2003 [Corrected by Carmine Suriano, Nov 08 2013]
a(100)=2574, a(1000)=250731 & a(10000)=25007302 which differs from Sum{i=4..104}A004524(i)=2625, Sum{i=4..1004}A004524(i)=251250 & Sum{i=4..10004}A004524(i)=25012500.  Robert G. Wilson v, Apr 18 2003


LINKS



FORMULA

Let f(n) = (1/4)*(n^22*zeta(1/2)*n) then we have a(n) = f(n) + O(1). More precisely we claim that for n >= 2 we have a(n) = floor(f(n)+c) where c > Max{a(n)f(n) : n>=1} = a(153)  f(153) = 1.032880076066608813953... and we believe we can take c = 1.033.  Benoit Cloitre, Sep 23 2012


EXAMPLE

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
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 AA054040 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


MATHEMATICA

f[n_] := Block[{k = 0, s = 0}, While[s < n, k++; s = N[s + 1/Sqrt[k], 50]]; k]; Table[f[n], {n, 1, 60}]


PROG

(PARI) a(n)=if(n<0, 0, t=1; z=1; while(z<n, t++; z=z+1/sqrt(t)); t) \\ Benoit Cloitre, Sep 23 2012


CROSSREFS

See A019529 for a different version.


KEYWORD

nonn


AUTHOR

Asher Auel (asher.auel(AT)reed.edu), Apr 13 2000


EXTENSIONS



STATUS

approved



