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

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

Offset

1,1

Comments

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

Links

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.

Crossrefs

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

Sequence in context: A017017 A297665 A309347 * A334964 A309351 A003615

Adjacent sequences: A273232 A273233 A273234 * A273236 A273237 A273238

Keyword

nonn,more

Author

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

Status

approved