cp's OEIS Frontend

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

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