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

3, 6, 12, 18, 24, 36, 48, 72, 96, 108

Offset: 1

Author

Vladimir Shevelev, May 13 2016

Keywords

• nonn
• fini
• full

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

But d(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 A273016, A273018.

Crossrefs

Cf. A067128, A065119 (the intersection of these two sequences is the present sequence). Cf. also A003586, A273016, A273018.

Programs

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 *)

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