A307327 Number of superabundant m in the interval p_k# <= m < p_(k+1)#, where p_i# = A002110(i).

1, 2, 3, 5, 6, 6, 5, 9, 8, 9, 8, 11, 12, 11, 11, 10, 12, 12, 11, 14, 15, 15, 16, 12, 14, 14, 15, 12, 12, 12, 12, 14, 13, 14, 12, 12, 14, 15, 16, 15, 15, 16, 18, 15, 17, 18, 18, 21, 22, 17, 15, 19, 17, 15, 16, 17, 16, 16, 17, 18, 18, 17, 17, 16, 17, 15, 15, 14

Offset: 0

Author

Michael De Vlieger, Apr 02 2019

Keywords

• nonn

Comments

Also first differences of the number of terms m in A004394 such that m < A002110(k).

Analogous to A307113.

Terms m in A004394 (superabundant numbers) are products of primorials.

The primorial A002110(k) is the smallest number that is the product of the k smallest primes.

This sequence partitions A004394 using terms in A002110.

First terms {1, 2, 3, 5, 6} are the same as those of A307113, since the first 19 terms of A002182 and A004394 are identical.

Examples

			First terms of this sequence and the superabundant numbers within the intervals:
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    6    2520    5040   10080   15120   25200   27720
6    5   55440  110880  166320  277200  332640
...
(Asterisks denote primorials in A004394.)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..407
• Michael De Vlieger, Graph comparing a(n) in red with A307113(n) in blue

Crossrefs

Cf. A002110, A002182, A004394, A307113.

Programs

Mathematica

Block[{nn = 8, P, s}, P = Nest[Append[#, #[[-1]] Prime@ Length@ #] &, {1}, nn + 1]; s = Array[DivisorSigma[1, # ]/# &, 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}]]