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A273235 Number of Ramanujan's largely composite numbers having prime(n) as the greatest prime divisor. 0
3, 10, 17, 28, 27, 43, 44, 69, 68, 58, 97, 97, 125, 164, 201, 185, 162, 254, 263, 313, 491, 434, 466, 417, 309, 358, 510, 633, 935, 1148, 454 (list; graph; refs; listen; history; text; internal format)



Theorem. The sequence is unbounded.

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).

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.

So we have a contradiction. This means that M_n and this sequence are unbound. QED


Table of n, a(n) for n=1..31.

P. Erdős, On Highly composite numbers, J. London Math. Soc. 19 (1944), 130--133 MR7,145d; Zentralblatt 61,79.


Cf. A067128, A273015, A273016, A273018, A273057.

Sequence in context: A017017 A297665 A309347 * A334964 A309351 A003615

Adjacent sequences:  A273232 A273233 A273234 * A273236 A273237 A273238




Vladimir Shevelev and Peter J. C. Moses, May 18 2016



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Last modified May 28 15:55 EDT 2020. Contains 334684 sequences. (Running on oeis4.)