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 A273015 Ramanujan's largely composite numbers having 3 as the greatest prime divisor. 6
 3, 6, 12, 18, 24, 36, 48, 72, 96, 108 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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_1-1 such that 2^x_1 > p_(k+1). Indeed, consider N_1 = 2^(l_1-x_1)*3^l_2*...*p_k^l_k*p_(k+1). Since 2^x_1 > p_(k+1) then N_1 d(N) if l_1 >= 2*x_1, so l_1 <= 2x_1-1. Analogously we find l_i <= 2x_i-1 if p_i^x_i > p_(k+1), i <= k. Therefore N <= 2^(2*x_1-1)*3^(2*x_2-1)*...* p_k^(2*x_k-1) 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 A273215, A273216, A273218. LINKS MATHEMATICA 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 *) CROSSREFS Cf. A067128, A065119 (the intersection of these two sequences is the present sequence). Cf. also A003586. Sequence in context: A160738 A028882 A154907 * A242297 A024513 A181026 Adjacent sequences:  A273012 A273013 A273014 * A273016 A273017 A273018 KEYWORD nonn,fini,full AUTHOR Vladimir Shevelev, May 13 2016 STATUS approved

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Last modified December 16 09:06 EST 2019. Contains 330020 sequences. (Running on oeis4.)