A305056 -id:A305056 - cp's OEIS frontend

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

Michael De Vlieger, Dec 29 2020

nonn

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.

			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.

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.

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

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

A340137 Numbers k in A305056 such that k*A002110(j) is in A004490.

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 144, 720, 1440, 10080, 30240, 60480, 302400, 3326400, 6652800, 19958400, 259459200, 518918400, 3632428800, 61751289600, 1173274502400, 3519823507200, 17599117536000, 35198235072000, 809559406656000, 1619118813312000, 46954445586048000

Offset: 1

Michael De Vlieger, Jan 08 2021

nonn

All terms are in A025487, since all terms m in A004490 are products of primorials P in A002110.

Let Q = A002110(A001221(m)) be the largest primorial divisor Q | m. The terms in this sequence are the primitive quotients k = m/Q for m in A004490.

			a(1) = 1 since there are 2 colossally abundant numbers m that are primorials P, i.e., 2 and 6.
a(2) = 2 since 2 colossally abundant numbers m = 2P, i.e., 12 and 60.
a(3) = 4 since 120 = 4*30 is colossally abundant.
a(4) = 12 since 360 and 2520 = 12P, etc.
Table showing products of primorials in the column heading and terms in this sequence in the row headings that appear in A004490 (and in these cases, also A002201, thereby in their intersection, A224078).
          2   6   30    210    2310    30030      510510
  ------------------------------------------------------
    1:    2   6
    2:       12   60
    4:           120
   12:           360   2520
   24:                 5040   55440   720720
   48:                               1441440
  144:                               4324320
  720:                              21621600   367567200   ...
Textual plot of numbers at (n,k) where row n = a(n) and column k = A002110(k), marking terms (x) in A224078, (*) only in A004490, or (.) only in A002201.
   1: xx
   2:  xx
   3:   x
   4:   xx
   5:    xxx
   6:      x
   7:      x
   8:      xxx*
   9:        .x**
  10:         ..*
  11:          .x***
  12:           ...xx**
  13:               ..x****
  14:                     **
  15:                 ..   **
  16:                  .....***
  17:                      ...**********
  18:                        .....     ***
  19:                            ...     ****
  20:                              .....    ********
The largest term in A224078 = 581442729886633902054768000 = a(13)*A002110(17), so appears at (13,17).

Michael De Vlieger, Table of n, a(n) for n = 1..144
Michael De Vlieger, Annotated color-coded plot (x,y) = (a(n), A002110(j)) highlighting colossally abundant numbers in red. This sequence also can portray many but not all superior highly composite numbers (shown in blue). Terms in A224078 appear in black.
Michael De Vlieger, Simple extended color-coded plot (x,y) = (a(n), A002110(m)) showing 1000 terms of A004490 in red.

Cf. A002110, A004490, A025487, A073751, A301416, A305056, A307327.

Block[{s = Import["https://oeis.org/A073751/b073751.txt", "Data"][[All, -1]], a = 1, b = {}, k, m = 0}, Do[k = a*s[[i]]; If[# > m, m++] &@ PrimePi@ s[[i]]; Set[a, k]; AppendTo[b, k/Product[Prime[j], {j, m}]], {i, 120}]; Union@ b]

cp's OEIS Frontend

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

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