[seqfan] Suggested method to generate A004080 b-file
David Wilson
dwilson at gambitcomm.com
Wed Jun 17 16:16:28 CEST 2009
For you math package freaks out there:
Equation (15) on http://mathworld.wolfram.com/HarmonicNumber.html
gives a very good estimate of H_k, the kth harmonic number, specifically
ln(k+1/2)+gamma+1/(24(k+1)^2) < H_k < ln(k+1/2)+gamma+1/(24k^2) [1]
where gamma is the Euler-Mascheroni constant, which is known very
accurately (to billions of digits).
A004080(n) is the smallest k with H_k > n. The interval in [1], which is
O(k^-3) in length, is extremely unlikely to straddle an integer for the
large k in question. It should be sufficient to decide whether a given
H_k is less than or greater than integer n. This is the basis of a
binary or adaptive search algorithm for A004080(n), which would help us
generate a b-file for that sequence.
A good test would be A004080(100)
= A099618(2)
= 15092688622113788323693563264538101449859497
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