cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A273235 Number of Ramanujan's largely composite numbers having prime(n) as the greatest prime divisor.

This page as a plain text file.
%I A273235 #17 May 21 2016 23:41:15
%S A273235 3,10,17,28,27,43,44,69,68,58,97,97,125,164,201,185,162,254,263,313,
%T A273235 491,434,466,417,309,358,510,633,935,1148,454
%N A273235 Number of Ramanujan's largely composite numbers having prime(n) as the greatest prime divisor.
%C A273235 Theorem. The sequence is unbounded.
%C A273235 Proof. Since the sequence of highly composite numbers (A002182) is a subsequence of this sequence, it is sufficient to prove that the number M_n of highly composite numbers with the maximal prime divisor p_n is unbounded. Let N be a large highly composite number. Then for the greatest prime divisor p_N of N we have [Erdos] p_N=O(log N). So for all N<=x, p_N=O(log x).
%C A273235 If M_n=O(1), then the number of all highly composite numbers <=x is O(p_n)=O(log x). However, Erdos [Erdos] proved that this number is more than (log x)^(1+c) for a certain c>0.
%C A273235 So we have a contradiction. This means that M_n and this sequence are unbound. QED
%H A273235 P. Erdős, <a href="https://www.renyi.hu/~p_erdos/1944-04.pdf">On Highly composite numbers</a>, J. London Math. Soc. 19 (1944), 130--133 MR7,145d; Zentralblatt 61,79.
%Y A273235 Cf. A067128, A273015, A273016, A273018, A273057.
%K A273235 nonn,more
%O A273235 1,1
%A A273235 _Vladimir Shevelev_ and _Peter J. C. Moses_, May 18 2016