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

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

Author

Michael De Vlieger, Jan 08 2021

Keywords

• nonn

Comments

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.

Examples

			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).
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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]