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Number of ordered pairs (i,j) with i>=1, j>=1 and i!*j! = n.
2

%I #16 Mar 06 2020 12:02:20

%S 0,1,2,0,1,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,

%T 0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Number of ordered pairs (i,j) with i>=1, j>=1 and i!*j! = n.

%H Eric M. Schmidt, <a href="/A108069/b108069.txt">Table of n, a(n) for n = 0..10000</a>

%H D. M. Kane, <a href="http://www.emis.de/journals/INTEGERS/papers/f2/f2.Abstract.html">On the number of ways of writing T as a product of factorials</a>, Integers: Electronic Journal of Combinatorial Number Theory, 5 (2005), #A02, 10pp.

%F Conjecture: max_{n->infinity} a(n) = 4 [Kane].

%e 1 = 1!*1!, so a(1) = 1; 2 = 1!*2! = 2!*1!, so a(2) = 2. - corrected by _Eric M. Schmidt_, Dec 07 2012

%o (Sage)

%o def A108069(n) :

%o count = 0;

%o m = 1; factm = 1;

%o while factm <= n :

%o k = 1; factk = 1;

%o while factm * factk <= n :

%o if factm * factk == n :

%o count += 1;

%o k += 1; factk *= k;

%o m += 1; factm *= m;

%o return count; # _Eric M. Schmidt_, Dec 07 2012

%Y Cf. A108068.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Jun 04 2005

%E More terms from _Christian G. Bower_, Jun 04 2005

%E a(0) term added by _Eric M. Schmidt_, Dec 07 2012