Theorem. Ramanujan's largely composite numbers (A067128) having the greatest prime divisor p_k = prime(k) do not exceed Product_{2 <= p <= p_k} p^((2*ceiling(log_p(p_(k + 1))  1).
Proof. Let N be in A067128 with prime power factorization 2^l_1 * 3^l_2 * ... * p_k^l_k.
First let us show that l_1 <= 2x_11 such that 2^x_1 > p_(k+1).
Indeed, consider N_1 = 2^(l_1x_1)*3^l_2*...*p_k^l_k*p_(k+1).
Since 2^x_1 > p_(k+1) then N_1<N.
But d(N_1) > d(N) if l_1 >= 2*x_1, so l_1 <= 2x_11.
Analogously we find l_i <= 2x_i1 if p_i^x_i > p_(k+1), i <= k.
Therefore N <= 2^(2*x_11)*3^(2*x_21)*...* p_k^(2*x_k1) and the theorem easily follows.
QED
The inequality of the theorem gives a way to find the full sequence for every p_k. In particular, in case p_k = 2 we have the sequence {2, 4, 8}. For other cases see A273016, A273018.
