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A340014 Numbers k in A305056 such that k * A002110(j) is in A004394 for some j >= 0.

%I A340014 #7 Jan 08 2021 16:59:02
%S A340014 1,2,4,6,8,12,24,48,72,120,144,240,288,360,720,1440,2160,2880,4320,
%T A340014 5040,8640,10080,15120,20160,30240,60480,120960,151200,181440,241920,
%U A340014 302400,604800,907200,1209600,1330560,1663200,1814400,3326400,6652800,9979200,13305600
%N A340014 Numbers k in A305056 such that k * A002110(j) is in A004394 for some j >= 0.
%C A340014 Let m be a superabundant number. Since m is a product of primorials P, we may identify a greatest primorial divisor P(omega(m)) = A002110(A001221(A004394(n))).
%C A340014 This sequence lists the primitive quotients k = m/P(omega(m)).
%C A340014 Since m is a product of primorials and k is the quotient resulting from division of m by the largest primorial divisor P, this sequence is also a subset of A025487, which in turn is a subset of A055932.
%C A340014 We can plot all m in A004394 at (A002110(j),k), but this sequence does not accommodate all highly composite numbers; it is missing k = {36, 96, 216, 480, ...}. In contrast, k in A301414 can represent all superabundant numbers m, but a(116)=592424239959167616000 is the least k missing. Therefore in order to plot both A002182 and A004394 one must use the union of a(n) and A301414(n). One can ably plot all the terms common to both A002182 and A004394 (i.e., A166981) using k in A301414.
%H A340014 Michael De Vlieger, <a href="/A340014/b340014.txt">Table of n, a(n) for n = 1..2000</a>
%H A340014 Michael De Vlieger, <a href="/A340014/a340014.png">Annotated color-coded plot</a> (x,y) = (a(n), A002110(j)) highlighting superabundant numbers.
%H A340014 Michael De Vlieger, <a href="/A340014/a340014_1.png">Extended annotated color-coded plot</a> showing all terms in A004394 that also appear in A224078.
%H A340014 Michael De Vlieger, <a href="/A340014/a340014_2.png">Simple extended color-coded plot</a> (x,y) = (a(n), A002110(m)) for coordinates between (1,0)..(225,379), with (1,0) in lower left corner.
%e A340014 Plot of (A002110(j),k) with k a term in this sequence such that A002110(j) * k is in A004394. Asterisks denote products that are in A004490.
%e A340014    {0,1} {1,1} {2,1}
%e A340014      1     2*    6*
%e A340014          {1,2} {2,2} {3,2}
%e A340014            4     12*   60*
%e A340014                {2,4} {3,4}  {4,4}
%e A340014                  24   120*   840
%e A340014                {2,6} {3,6}  {4,6}
%e A340014                  36   180    1260
%e A340014                {2,8} {3,8}  {4,8}
%e A340014                  48   240    1680
%e A340014                     {3,12} {4,12}   {5,12}
%e A340014                       360*   2520*   27720
%e A340014                     {3,24} {4,24}   {5,24}    {6,24}
%e A340014                       720    5040*   55440*   720720*
%e A340014                            {4,48}   {5,48}    {6,48}
%e A340014                             10080   110880   1441440*
%e A340014                             ...     ...      ...       ...
%e A340014 This table is missing 7560, 83160, 1081080 at {4,36}, {5,36}, and {6,36}, respectively, which are numbers in A002182 but not in A004394. Thus, 36 is in A301414 but not in this sequence.
%t A340014 Block[{s = Array[DivisorSigma[1, #]/# &, 10^6], t}, t = Union@ FoldList[Max, s]; Union@ Map[#/Product[Prime@ i, {i, PrimeNu@ #}] &@ First@ FirstPosition[s, #] &, t]]
%Y A340014 Cf. A002110, A004394, A004490, A025487, A055932, A301414, A305056.
%K A340014 nonn
%O A340014 1,2
%A A340014 _Michael De Vlieger_, Dec 29 2020