%I #12 Mar 14 2017 00:34:17
%S 1,4,7,13,13,25,19,38,37,49,31,74,37,73,79,104,49,125,55,143,121,121,
%T 67,203,121,145,171,221,85,263,91,272,205,193,199,354,109,217,247,383,
%U 121,398,127,377,381,265,139,531,253,443,331,455,157,558,355,587,373
%N Number of integer triples (x,y,z), x > 0, y > 0, z > 0, such that x <= p, y <= q, z <= r for any factorization n = p*q*r.
%C This sequence is a three-dimensional equivalent of A138808.
%C a(n) >= 3*n-2, with equality iff n is not composite.
%C a(n) >= 3*A138808(n)-3*n+1, with equality iff n has at most 2 prime factors (counted with multiplicity).
%H Rémy Sigrist, <a href="/A283626/b283626.txt">Table of n, a(n) for n = 1..1000</a>
%H Rémy Sigrist, <a href="/A283626/a283626.png">Illustration of the first terms</a>
%o (PARI) a(n)=my(h=matrix(n,n)); fordiv(n,d, fordiv(n/d,dd, for(x=1,d, for(y=1,dd,h[x,y]=max(h[x,y],n/d/dd))))); return(sum(x=1,n, sum(y=1,n,h[x,y])))
%Y Cf. A138808.
%K nonn
%O 1,2
%A _Rémy Sigrist_, Mar 12 2017