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

1, 2, 4, 6, 8, 12, 24, 48, 72, 120, 144, 240, 288, 360, 720, 1440, 2160, 2880, 4320, 5040, 8640, 10080, 15120, 20160, 30240, 60480, 120960, 151200, 181440, 241920, 302400, 604800, 907200, 1209600, 1330560, 1663200, 1814400, 3326400, 6652800, 9979200, 13305600

Offset

1,2

Comments

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))).

This sequence lists the primitive quotients k = m/P(omega(m)).

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.

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..2000

Michael De Vlieger, Annotated color-coded plot (x,y) = (a(n), A002110(j)) highlighting superabundant numbers.

Michael De Vlieger, Extended annotated color-coded plot showing all terms in A004394 that also appear in A224078.

Michael De Vlieger, Simple extended color-coded plot (x,y) = (a(n), A002110(m)) for coordinates between (1,0)..(225,379), with (1,0) in lower left corner.

Example

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.
   {0,1} {1,1} {2,1}
     1     2*    6*
         {1,2} {2,2} {3,2}
           4     12*   60*
               {2,4} {3,4}  {4,4}
                 24   120*   840
               {2,6} {3,6}  {4,6}
                 36   180    1260
               {2,8} {3,8}  {4,8}
                 48   240    1680
                    {3,12} {4,12}   {5,12}
                      360*   2520*   27720
                    {3,24} {4,24}   {5,24}    {6,24}
                      720    5040*   55440*   720720*
                           {4,48}   {5,48}    {6,48}
                            10080   110880   1441440*
                            ...     ...      ...       ...
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.

Mathematica

Block[{s = Array[DivisorSigma[1, #]/# &, 10^6], t}, t = Union@ FoldList[Max, s]; Union@ Map[#/Product[Prime@ i, {i, PrimeNu@ #}] &@ First@ FirstPosition[s, #] &, t]]

Crossrefs

Cf. A002110, A004394, A004490, A025487, A055932, A301414, A305056.

Sequence in context: A333953 A333963 A307187 * A001217 A131885 A173941

Adjacent sequences: A340011 A340012 A340013 * A340015 A340016 A340017

Keyword

nonn

Author

Michael De Vlieger, Dec 29 2020

Status

approved