A307113 Number of highly composite numbers (m in A002182) in the interval p_k# <= m < p_(k+1)#, where p_i# = A002110(i).

1, 2, 3, 5, 6, 8, 10, 12, 13, 15, 14, 15, 17, 16, 16, 19, 17, 21, 19, 20, 26, 22, 25, 26, 25, 29, 28, 26, 27, 28, 29, 33, 33, 34, 37, 37, 35, 35, 39, 37, 38, 38, 37, 37, 38, 38, 41, 38, 37, 36, 37, 37, 40, 44, 44, 45, 44, 44, 45, 45, 49, 48, 52, 51, 53, 52, 51

Offset: 0

Author

Michael De Vlieger, Mar 25 2019

Keywords

• nonn

Comments

Terms m in A002182 (highly composite numbers or HCNs) are products of primes p <= q, where q is the greatest prime factor of m. The primorial A002110(k) is the smallest number that is the product of the k smallest primes. This sequence partitions A002182 using terms in A002110.

Examples

			a(3) = 5 since there are 5 highly composite numbers A002110(3) <= m < A002110(4), i.e., 30 <= m < 210: {36, 48, 60, 120, 180}.
n  a(n)      m such that A002110(n) <= m < A002110(n+1)
--------------------------------------------------------------------
0    1       1
1    2       2      4
2    3       6     12     24
3    5      36     48     60     120     180
4    6     240    360    720     840    1260    1680
5    8    2520   5040   7560   10080   15120   20160   25200   27720
...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..4149
• A. Flammenkamp, List of the first 779,674 highly composite numbers

Crossrefs

Cf. A002110, A002182.

Programs

Mathematica

Block[{nn = 8, P, s}, P = Nest[Append[#, #[[-1]] Prime@ Length@ #] &, {1}, nn + 1]; s = DivisorSigma[0, Range@ P[[nn + 1]] ]; s = Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]; Table[Count[s, _?(If[! IntegerQ@ #, 1, #] &@ P[[i]] <= # < P[[i + 1]] &)], {i, nn}]]