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A273015 Ramanujan's largely composite numbers having 3 as the greatest prime divisor.

%I A273015 #29 Oct 31 2020 11:50:52
%S A273015 3,6,12,18,24,36,48,72,96,108
%N A273015 Ramanujan's largely composite numbers having 3 as the greatest prime divisor.
%C A273015 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).
%C A273015 Proof. Let N be in A067128 with prime power factorization 2^l_1 * 3^l_2 * ... * p_k^l_k.
%C A273015 First let us show that l_1 <= 2x_1-1 such that 2^x_1 > p_(k+1).
%C A273015 Indeed, consider N_1 = 2^(l_1-x_1)*3^l_2*...*p_k^l_k*p_(k+1).
%C A273015 Since 2^x_1 > p_(k+1) then N_1<N.
%C A273015 But d(N_1) > d(N) if l_1 >= 2*x_1, so l_1 <= 2x_1-1.
%C A273015 Analogously we find l_i <= 2x_i-1 if p_i^x_i > p_(k+1), i <= k.
%C A273015 Therefore N <= 2^(2*x_1-1)*3^(2*x_2-1)*...* p_k^(2*x_k-1) and the theorem easily follows.
%C A273015 QED
%C A273015 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.
%t A273015 a = {}; b = {0}; Do[If[# >= Max@ b, AppendTo[a, k] && AppendTo[b, #]] &@ DivisorSigma[0, k], {k, 10^7}]; Select[a, FactorInteger[#][[-1, 1]] == 3 &] (* _Michael De Vlieger_, May 13 2016 *)
%Y A273015 Cf. A067128, A065119 (the intersection of these two sequences is the present sequence). Cf. also A003586, A273016, A273018.
%K A273015 nonn,fini,full
%O A273015 1,1
%A A273015 _Vladimir Shevelev_, May 13 2016