%I #8 Apr 19 2019 13:49:11
%S 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,
%T 15,12,12,12,12,14,13,14,12,12,14,15,16,15,15,16,18,15,17,18,18,21,22,
%U 17,15,19,17,15,16,17,16,16,17,18,18,17,17,16,17,15,15,14
%N Number of superabundant m in the interval p_k# <= m < p_(k+1)#, where p_i# = A002110(i).
%C Also first differences of the number of terms m in A004394 such that m < A002110(k).
%C Analogous to A307113.
%C Terms m in A004394 (superabundant numbers) are products of primorials.
%C The primorial A002110(k) is the smallest number that is the product of the k smallest primes.
%C This sequence partitions A004394 using terms in A002110.
%C 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.
%H Michael De Vlieger, <a href="/A307327/b307327.txt">Table of n, a(n) for n = 0..407</a>
%H Michael De Vlieger, <a href="/A307327/a307327.png">Graph comparing a(n) in red with A307113(n) in blue</a>
%e First terms of this sequence and the superabundant numbers within the intervals:
%e n a(n) m such that A002110(n) <= m < A002110(n+1)
%e -------------------------------------------------------
%e 0 1 1*
%e 1 2 2* 4
%e 2 3 6* 12 24
%e 3 5 36 48 60 120 180
%e 4 6 240 360 720 840 1260 1680
%e 5 6 2520 5040 10080 15120 25200 27720
%e 6 5 55440 110880 166320 277200 332640
%e ...
%e (Asterisks denote primorials in A004394.)
%t 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}]]
%Y Cf. A002110, A002182, A004394, A307113.
%K nonn
%O 0,2
%A _Michael De Vlieger_, Apr 02 2019