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

0,2

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.

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

Example

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.)

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}]]

Crossrefs

Cf. A002110, A002182, A004394, A307113.

Sequence in context: A175578 A347861 A316609 * A000036 A165081 A165089

Adjacent sequences: A307324 A307325 A307326 * A307328 A307329 A307330

Keyword

nonn

Author

Michael De Vlieger, Apr 02 2019

Status

approved