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%I #8 Jan 12 2021 18:03:39
%S 1,2,4,12,24,48,144,720,1440,10080,30240,60480,302400,3326400,6652800,
%T 19958400,259459200,518918400,3632428800,61751289600,1173274502400,
%U 3519823507200,17599117536000,35198235072000,809559406656000,1619118813312000,46954445586048000
%N Numbers k in A305056 such that k*A002110(j) is in A004490.
%C All terms are in A025487, since all terms m in A004490 are products of primorials P in A002110.
%C 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.
%H Michael De Vlieger, <a href="/A340137/b340137.txt">Table of n, a(n) for n = 1..144</a>
%H Michael De Vlieger, <a href="/A340137/a340137.png">Annotated color-coded plot</a> (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.
%H Michael De Vlieger, <a href="/A340137/a340137_1.png">Simple extended color-coded plot</a> (x,y) = (a(n), A002110(m)) showing 1000 terms of A004490 in red.
%e a(1) = 1 since there are 2 colossally abundant numbers m that are primorials P, i.e., 2 and 6.
%e a(2) = 2 since 2 colossally abundant numbers m = 2P, i.e., 12 and 60.
%e a(3) = 4 since 120 = 4*30 is colossally abundant.
%e a(4) = 12 since 360 and 2520 = 12P, etc.
%e 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).
%e 2 6 30 210 2310 30030 510510
%e ------------------------------------------------------
%e 1: 2 6
%e 2: 12 60
%e 4: 120
%e 12: 360 2520
%e 24: 5040 55440 720720
%e 48: 1441440
%e 144: 4324320
%e 720: 21621600 367567200 ...
%e 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.
%e 1: xx
%e 2: xx
%e 3: x
%e 4: xx
%e 5: xxx
%e 6: x
%e 7: x
%e 8: xxx*
%e 9: .x**
%e 10: ..*
%e 11: .x***
%e 12: ...xx**
%e 13: ..x****
%e 14: **
%e 15: .. **
%e 16: .....***
%e 17: ...**********
%e 18: ..... ***
%e 19: ... ****
%e 20: ..... ********
%e The largest term in A224078 = 581442729886633902054768000 = a(13)*A002110(17), so appears at (13,17).
%t 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]
%Y Cf. A002110, A004490, A025487, A073751, A301416, A305056, A307327.
%K nonn
%O 1,2
%A _Michael De Vlieger_, Jan 08 2021
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