OFFSET
0,2
COMMENTS
Number of factorizations of (p*q*r)^n into distinct factors where p, q, r are distinct primes.
FORMULA
a(n) = [(x*y*z)^n] 1/2 * Product_{i,j,k>=0} (1+x^i*y^j*z^k).
EXAMPLE
a(0) = 1: [].
a(1) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(0,1,1),(1,0,0)], [(0,0,1),(0,1,0),(1,0,0)].
MAPLE
with(numtheory):
b:= proc(n, k) option remember;
`if`(n>k, 0, 1) +`if`(isprime(n), 0,
add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
end:
a:= n-> b(30^n$2):
seq(a(n), n=0..10); # Alois P. Heinz, May 26 2013
MATHEMATICA
b[n_, k_] := b[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, b[n/d, d - 1]], {d, Divisors[n][[2 ;; -2]]}]]; a[0] = 1; a[n_] := b[30^n, 30^n]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Alois P. Heinz, Nov 23 2012
EXTENSIONS
a(16) from Alois P. Heinz, May 26 2013
a(17) from Alois P. Heinz, Sep 24 2014
More terms from Jean-François Alcover, Jan 15 2016
STATUS
approved