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

Author

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

Keywords

• nonn
• more

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

• 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.