I always wondered how fast the denominators of the harmonic numbers grow. This is answered in: http://mathworld.wolfram.com/HarmonicNumber.html It says that for n = 0, 1, 2, 3, ..., the number of digits in the denominator of HarmonicNumber[10^n] is given by 1, 4, 40, 433, 4345, 43450, 434110, 4342302, 43428678, ... (Sloane's A114468). These digits converge to what appears to be the decimal digits of log_(10)e==0.43429448... I don't know about anyone else, but I think this is neat! (Is it obvious?) Bob --- franktaw@netscape.net wrote:
The numerators are A001008, and the denominators are A002805.
The denominators are the easy part, so I'll mostly address that.
The q(n)'s definitely have atypical factorizations; in particular, q(n) divides n!. (In fact, it divides lcm(1,2,3,...,n).) Note that if p is prime, q(p^k) is always divisible by p^k (in fact, q(n) is divisible by p^k for p^k <= n < 2*p^k - you can improve this result further for p != 3).
The numerators are also divisible by every p (except 2): see http://mathworld.wolfram.com/WolstenholmesTheorem.html/.
Franklin T. Adams-Watters
-----Original Message----- From: dasimov@earthlink.net
The harmonic number H(n) := 1 + 1/2 + 1/3 + . . . + 1/n has the interesting property that although it -> oo as n does, it's never an integer.
So let H(n) = p(n)/q(n) in lowest terms. (I have full confidence that p and q are already in the OEIS.)
What is known about the factorizations of the p(n)'s and q(n)'s ?
In particular, are their factorizations known or believed to have statistical properties atypical of numbers of the same size (re the number, size, and exponents of their prime factors) ?
Do the factors of p(n) (or q(n)) have special number-theoretical properties? (I.e., what is known about the primes -- if any -- that *never* occur as a factor of any p(n) (q(n)) ?
--Dan
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