[seqfan] Re: g(k) = (k^2)!*prod(j=0, k-1, j!/(j+k)!)
wouter meeussen
wouter.meeussen at pandora.be
Mon Mar 1 18:53:05 CET 2010
hi Chris,
define w[n, b] as (n - sum_of_digits_of_n_in_base_b)/(b-1)
then w[n, p] fro p prime, equals the power of p in the prime decomposition
of (n!)
since you seek the prime exponents of an expression consisting of a product
of factorials,
you just need to properly sum the prime exponents
w[n^2, p]+ Sum[k=0...n-1 ; w[k,p]-w[n+k,p] ]
for any n an p. See?
Wouter.
> for some of us who suffer from the Mathematica dialect:
> Table[b=Prime[p];
> w[n^2]+ Sum[w[k]-w[n+k] ,{k,0,n-1}]/.
> w[q_]:>(q-Plus@@IntegerDigits[q,b])/(b-1),{p,22},{n,22}]
>
> is exactly the table you showed.
> Don't count the trailing zero's of n! in base b, but count the carries in
n.
> I remember this being called 'minor numiracle', big tongue in cheek.
>
> guess you knew this already.
>
> Wouter.
>
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