A301414 -id:A301414 - cp's OEIS frontend

A301413 a(n) = A002182(n)/A002110(A108602(n)).

1, 1, 2, 1, 2, 4, 6, 8, 2, 4, 6, 8, 12, 24, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 12, 216, 240, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, 72, 1440, 120, 144, 216, 240, 288, 360, 480, 576

Offset: 1

Michael De Vlieger, Mar 30 2018

nonn

This sequence appears in Siano paper, page 5 of 12, as the "variable part" v. - Michael De Vlieger, Oct 11 2023

			Let m be a value in this sequence. The table below shows m*A002110(A108602(k)). Columns are A108602(k), rows are m whose products m*A002110(A108602(k)) appear in A002182 are in this sequence. Numbers in A002182 that also appear in A002201 are followed by (*).
        0  1   2    3     4       5       6 ...
      +------------------------------------
    1 | 1* 2*  6*
    2 |    4  12*  60*
    4 |       24  120*  840
    6 |       36  180  1260
    8 |       48  240  1680
   12 |           360* 2520*  27720
   24 |           720  5040*  55440* 720720*
   ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, On a graph of highly composite numbers
A. Flammenkamp, Highly composite numbers
D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite Numbers, 1994.

Cf. A002110, A002182, A108602, A301414.

(* Load b-file from A002182 *)
With[{s = Import["b002182.txt","Data"][[All,-1]]}, Array[#/Product[Prime@ i, {i, PrimeNu[#]}] &@ s[[#]] &, 62]]

a(n) = A002182(n)/A007947(A002182(n)).

A059992 Numbers with an increasing number of nonprime divisors.

Original entry on oeis.org

1, 4, 8, 12, 24, 36, 48, 60, 72, 120, 180, 240, 360, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 45360, 50400, 55440, 75600, 83160, 110880, 151200, 166320, 221760, 277200, 332640

Offset: 1

Robert G. Wilson v, Mar 08 2001

nonn

Positions of records in A033273.
From Michael De Vlieger, Jan 04 2025: (Start)
Conjecture: This sequence includes all highly composite numbers (from A002182) except 2 and 6, but there are other terms in this sequence (e.g., a(3) = 8, a(9) = 72) that are not highly composite.
Conjecture: a(n)/A007947(a(n)) is in A301414. (End)

			a(4)=12 because twelve has 4 nonprime divisors {1, 4, 6 and 12} whereas a(3)=8 has only 3; and twelve is the first number greater than eight which exhibits this property.

Amiram Eldar, Table of n, a(n) for n = 1..571 (terms 1..146 from Ray Chandler)
Michael De Vlieger, Plot S(n) = P(omega(n))*m at (x,y) = (m, omega(n)), where S is the union of A002182 and this sequence, P is A002110, omega is A001221, and only select m that harbor S(n) shown. Shows the coincidence of many terms in this sequence with A002182. Blue represents m in A002182, gold m in both A002182 and this sequence; dark blue represents m in A002201 (and also in A002182), orange m in both A002201 and this sequence; red indicates terms in this sequence that are not in A002182. Green highlights terms in A002182 but are not determined to be in this sequence.

Cf. A055079, A002182, A033273, A180040.

l = 0; Do[ c = Count[PrimeQ[ Divisors[n] ], False]; If[c > l, l = c; Print[n] ], {n, 1, 10^6} ]

lista(nn) = {my(m=0, nb); for (n=1, nn, nb = sumdiv(n, d, !isprime(d)); if (nb > m, m = nb; print1(n, ", ")););} \\ Michel Marcus, Jul 16 2019

Alternate description and b-file from Ray Chandler, Aug 07 2010

A304234 Superior highly composite numbers that are superabundant but not colossally abundant.

Original entry on oeis.org

13967553600, 2248776129600, 65214507758400, 195643523275200, 12129898443062400, 448806242393308800, 18401055938125660800, 185942670254759802384000, 9854961523502269526352000, 1162885459773267804109536000, 780296143507862696557498656000

Offset: 1

Michael De Vlieger, May 08 2018

Numbers m in A002201 that are also in A004394 but not A004490.
Subset of A166981. Numbers in this sequence are in neither A224078 nor A304235.
There are 39 terms in this sequence.
The smallest term is 2^5 * 3^2 * 5 * A002110(8) or the product of A002110(k) with k = {1,1,1,2,3,8}.
The largest is 2^10 * 3^6 * 5^3 * 7^2 * 11 * 13 * 17 * 19 * 23 * A002110(65) or the product of A002110(k) with k = {1,1,1,1,2,2,2,3,4,9,65}, a 144 digit decimal number.

Michael De Vlieger, Table of n, a(n) for n = 1..39
Michael De Vlieger, Superior highly composite numbers m that are also superabundant but not colossally abundant
Michael De Vlieger, Color coded plot of m A002182 and A004394 at (x,y) where A301414(x) * A002110(y) = m, terms in this sequence are colored dark blue.
Michael De Vlieger, Annotated plot of a(n) for 1 <= n <= 39 at (x,y) = (a(n)/A002110(A001221(a(n)), A002110(A001221(a(n)))

Cf. A002110, A002182, A002201, A004394, A004490, A166981, A224078, A304235.

(* First, download b-files at A002201, A004394, and A004490 *)
f[w_] := Times @@ Flatten@ {Complement[#1, Union[#2, #3]], Product[Prime@ i, {i, PrimePi@ #}] & /@ #2, Factorial /@ #3} & @@ ToExpression@ {StringSplit[w, _?(! DigitQ@ # &)], StringCases[w, (x : DigitCharacter ..) ~~ "#" :> x], StringCases[w, (x : DigitCharacter ..) ~~ "!" :> x]};
With[{s = Import["b002201.txt", "Data"][[All, -1]], t = Select[Map[Which[StringTake[#, 1] == {"#"}, f@ Last@ StringSplit@ Last@ #, StringTake[#, 1] == {}, Nothing, True, ToExpression@ StringSplit[#][[1, -1]]] &, Drop[Import["b004394.txt", "Data"], 3] ], IntegerQ@ First@ # &][[All, -1]], u = Import["b004490.txt", "Data"][[All, -1]]}, Select[Intersection[s, t], FreeQ[u, #] &]]

A304235 Colossally abundant numbers that are highly composite, but not superior highly composite.

Original entry on oeis.org

160626866400, 9316358251200, 288807105787200, 2021649740510400, 224403121196654400, 9200527969062830400, 395622702669701707200, 1970992304700453905270400, 35468006523084668025340848000, 135483209545341953934626770390608000

Offset: 1

Michael De Vlieger, May 08 2018

Numbers m in A004490 that are also in A002182 but not A002201.
Subset of A166981. Numbers in this sequence are in neither A224078 nor A304234.
There are 32 terms in this sequence.
The smallest term is 2^4 * 3^2 * 5 * A002110(9) or the product of k = {1,1,2,3,9} in A002110.
The largest term is 2^9 * 3^5 * 5^3 * 7^2 * 11 * 13 * 17 * 19 * 23 * A002110(66) or the product of A002110(k) with k = {1,1,1,1,2,2,3,4,9,66}, a 146 digit decimal number.

Michael De Vlieger, Table of n, a(n) for n = 1..32
Michael De Vlieger, Colossally abundant m that are also highly composite but not superior highly composite
Michael De Vlieger, Color coded plot of m A002182 and A004394 at (x,y) where A301414(x) * A002110(y) = m, terms in this sequence are colored dark red.
Michael De Vlieger, Annotated plot of a(n) for 1 <= n <= 32 at (x,y) = (a(n)/A002110(A001221(a(n)), A002110(A001221(a(n)))

Cf. A002110, A002182, A002201, A004394, A004490, A166981, A224078, A304234.

(* First, download b-files at A002182, A002201, and A004490 *)
With[{s = Import["b004490.txt", "Data"][[All, -1]], t = Import["b002182.txt", "Data"][[All, -1]], u = Import["b002201.txt", "Data"][[All, -1]]}, Select[Intersection[s, t], FreeQ[u, #] &]]

A301415 Number of terms m in A002110 such that A301413(k) * A002110(m) is in A002182.

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 4, 3, 3, 5, 3, 4, 4, 5, 5, 5, 4, 3, 4, 4, 4, 6, 3, 4, 5, 4, 3, 4, 3, 7, 5, 5, 6, 9, 6, 5, 8, 6, 8, 8, 8, 6, 6, 8, 6, 5, 7, 8, 9, 5, 5, 7, 6, 5, 6, 5, 6, 5, 6, 9, 9, 6, 9, 9, 6, 6, 7, 8, 7, 7, 7, 9, 5, 10, 10, 5, 13, 9, 9, 8, 10, 10, 7, 10, 8

Offset: 1

Michael De Vlieger, Apr 09 2018

nonn

Numbers m = A301414(x) * A002110(y) that are in A002182 are plotted below. Those also in A002201 are followed by asterisk.
This sequence counts the terms in each column.
      1     2     3     4     5     6      7 ...
  +-----------------------------------------
0 |   1
1 |   2*    4
2 |   6*   12*   24    36    48
3 |        60*  120*  180   240   360*   720
4 |             840  1260  1680  2520*  5040*
5 |                             27720  55440*
6 |                                   720720*
...

			a(1) = 3 since A301414(1) = 1 produces 3 highly composite numbers when multiplied by primorials p_0#, p_1#, and p_2# = {1, 2, 6}.
a(2) = 3 since A301414(2) = 2 yields 3 HCNs, multiplied by p_1#, p_2#, and p_3# = {4, 12, 60}.

Michael De Vlieger, Table of n, a(n) for n = 1..8000
Michael De Vlieger, Plot m = A301414(x) * A002110(y) at (x,y) for all 779674 m in Achim Flammenkamp's dataset SHCNs are shown in red; 8563 * 4096 pixels.

Cf. A002110, A002182, A301413, A301414.

f[n_] := With[{d = FactorInteger@ n}, If[n == 1, {0}, ReplacePart[Table[0, {PrimePi[d[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, d]]]; Take[#, 85] &@ Block[{s = a002182, a, b, c, m, u}, s = Take[s, 1000]; a = Array[{#2, #1, StringTrim[StringReplace[ToString@ #, ", " -> "."], ("{" | "}") ...] &[#3 /. {} -> 0], Times @@ MapIndexed[Prime[First@ #2]^#1 &, #3]} & @@ {#1, Boole[First@ #2 > 0] Length@ #2, DeleteCases[-1 + #2, 0] /. -1 -> 0} & @@ {s[[#]], f@ s[[#]]} &, Length@ s]; u = Union@ a[[All, -1]]; b = MapIndexed[{i_, j_, k_, #1} -> ToExpression@ StringJoin["{i,", ToString@ First@ #2, ",", " j, k}"] &, Union@ a[[All, -1]]]; c = Map[# /. b &, a]; m = Max[c[[All, 2]] ]; c = Map[Sort@ # &, SplitBy[SortBy[c, First], First]]; Total /@ Transpose@ Array[With[{t = ConstantArray[0, m]}, ReplacePart[t, Map[#2 -> 1 & @@ # &, c[[#]] ] ] ] &, Length@ c] ]

A301416 Numbers k in A301413 such that k * A002110 (m) is in A002201.

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 144, 720, 1440, 10080, 30240, 60480, 302400, 604800, 6652800, 19958400, 259459200, 518918400, 3632428800, 61751289600, 185253868800, 926269344000, 17599117536000, 35198235072000, 809559406656000, 1619118813312000, 4857356439936000

Offset: 1

Michael De Vlieger, Apr 09 2018

nonn

			From _Michael De Vlieger_, May 14 2018: (Start)
Numbers m = A301416(x) * A002110(y) that are in A002201 are plotted below.
      1     2     3     4      5        6        7 ...
  +-----------------------------------------------
1 |   2
2 |   6    12
3 |        60   120   360
4 |                  2520   5040
5 |                        55440
6 |                       720720  1441440  4324320
...
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..170
Michael De Vlieger, Plot m = a(x) * A002110(y) at (x,y) for m in A002201, smallest 4096 terms m.

Cf. A002110, A002201, A301413, A301414.

t = Import["b002201.txt", "Data"][[All, -1]]; (* Uses b-file at A002201 Alternatively, use this conversion of terms at A000705 to a 10^4 term dataset for A002201. Processing 10^4 terms will take a long time: *) t = With[{s = Import["b000705.txt", "Data"][[All, -1]]}, FoldList[Times, s]]; f[n_] := With[{d = FactorInteger@ n}, If[n == 1, {0}, ReplacePart[Table[0, {PrimePi[d[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, d]]]; Union@ Array[Times @@ MapIndexed[Prime[First@#2]^#1 &, #3] & @@ {#1, Boole[First@ #2 > 0] Length@ #2, DeleteCases[-1 + #2, 0] /. -1 -> 0} & @@ {t[[#]], f@ t[[#]]} &, Length@ t]

A340840 Union of the highly composite and superabundant numbers.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880

Offset: 1

Michael De Vlieger, Jan 27 2021

nonn

Numbers m that set records in A000005 and numbers k that set records for the ratio A000203(k)/k, sorted, with duplicates removed.
All terms are in A025487, since all terms in A002182 and A004394 are products of primorials P in A002110.
For numbers that are highly composite but not superabundant, see A308913; for numbers that are superabundant but not highly composite, see A166735. - Jon E. Schoenfield, Jun 14 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated scatterplot of a(n) = A301414(x) * A002110(y) at (x, y) showing the intersection A166981 of A002182 and A004394, and the intersection A224078 of A002201 and A004490.

Cf. A000005, A000203, A027750, A002110.
Supersets: A025487, A055932.
Subsets: A002182, A002201, A004394, A004490, A166735, A166981, A168263, A189228, A224078, A304234, A304235, A308913, A333655, A338786.

(* Load the function f[] at A025487, then: *) Block[{t = Union@ Flatten@ f[15], a = {}, b = {}, d = 0, s = 0}, Do[(If[#2 > d, d = #2; AppendTo[a, #1]]; If[#3/#1 > s, s = #3/#1; AppendTo[b, #1]]) & @@ Flatten@ {t[[i]], DivisorSigma[{0, 1}, t[[i]]]}, {i, Length@ t}]; Union[a, b]]

A365900 Highly composite numbers k that remain highly composite when recursively divided by squarefree kernel.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 60, 120, 180, 360, 720, 840, 1260, 2520, 5040, 7560, 25200, 27720, 55440, 83160, 277200, 720720, 1081080, 3603600, 10810800, 21621600, 61261200, 183783600, 367567200, 3491888400, 6983776800, 48886437600, 73329656400, 80313433200, 160626866400, 1124388064800, 1686582097200, 32607253879200, 48910880818800, 1010824870255200, 1516237305382800

Offset: 1

Michael De Vlieger, Oct 06 2023

Let h(n) = A002182(n).
Let f(x) = x/rad(x) = A301413(x), where rad(n) = A007947(n) is a primorial and x is in h.
If f(h(k)) = m is highly composite, then we apply f(m) until we reach 1 or m that is not highly composite.
Let S be the chain of highly composite terms that result from the recursion of f beginning with k in A002182. Terms in S are nondecreasing and each appear in this sequence. Example: beginning with h(51), we have {21621600, 720, 24, 4, 2, 1}. Terms that follow h(51) in the chain appear in this sequence.
There are 19 known terms j in S = A301414 = union({A301413}) that are highly composite. f(h(k)) = j is a necessary but insufficient condition for h(k) to appear in this sequence.
The numbers j in {48, 240, 10080, 15120, 20160, 50400, 17297280} do not yield terms in this sequence, because {48, 240, 10080, 50400} settle to 8, S(32) = h(22) = 15120 settles to 72, S(33) = h(23) = 20160 ends up at 96, and the largest of the 19 terms, S(62) = h(50) = 17297280 ends up at 576, all of which are not highly composite. It appears that there are only 19 terms that enable membership in this sequence.

			1 is in this sequence since f(1) = 1 and 1 is highly composite.
2 is in this sequence since f(2) = 1 and 1 is highly composite.
12 is in this sequence since f(12) = 2, and f(2) = 1, both highly composite.
48 is not in this sequence since f(48) = 48/6 = 8, and 8 is not highly composite.
Applying f recursively to h(128) = 1516237305382800 yields the following chain:
1516237305382800 -> 7560 -> 36 -> 6 -> 1, all highly composite. It seems that this is the largest term in the sequence.
.
Tree plot of terms:
1 --- 2 --- 4 --- 24 --- 720 --- 21621600
   |     |     |      |       |- 367567200
   |     |     |      |       |- 6983776800
   |     |     |      |       |_ 160626866400
   |     |     |      |
   |     |     |      |- 5040 -- 48886437600
   |     |     |      |       |- 1124388064800
   |     |     |      |       |- 32607253879200
   |     |     |      |       |_ 1010824870255200
   |     |     |      |
   |     |     |      |- 55440
   |     |     |      |_ 720720
   |     |     |
   |     |     |- 120 -- 25200
   |     |     |      |- 277200
   |     |     |      |- 3603600
   |     |     |      |_ 61261200
   |     |     |
   |     |     |_ 840
   |     |
   |     |-12 --- 360 -- 10810800
   |     |     |      |- 183783600
   |     |     |      |- 3491888400
   |     |     |      |_ 80313433200
   |     |     |
   |     |     |- 2520
   |     |     |_ 27720
   |     |
   |     |_60
   |
   |_ 6 -- 36 --- 7560 --- 73329656400
         |     |        |- 1686582097200
         |     |        |- 48910880818800
         |     |        |_ 1516237305382800
         |     |
         |     |_ 83160 -- 1081080
         |
         |-180
         |_1260

Michael De Vlieger, Plot of h(k) = S(i)*P(omega(h(n))) at (x,y) = (i, omega(h(k)) highlighting k such that h(k) is in this sequence.
Achim Flammenkamp, Highly composite numbers.
Eric Weisstein's World of Mathematics, Highly Composite Number.

Cf. A002110, A002182, A301413, A301414.

(* Program loads highly composite numbers from A002182 bfile *)
a2182 = Import["https://oeis.org/A002182/b002182.txt", "Data"][[All, -1]];
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Select[Array[
  NestWhileList[#/rad[#] &, a2182[[#]], And[# > 1, ! FreeQ[a2182, #]] &] &, 250],
  Last[#] == 1 &][[All, 1]]

A374907 Number whose divisors have a mean number of divisors that attains a record value.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, 1440, 2160, 2880, 4320, 5040, 7200, 7560, 8640, 10080, 14400, 15120, 20160, 30240, 40320, 50400, 60480, 90720, 100800, 120960, 151200, 181440, 241920, 302400, 362880, 453600

Offset: 1

Amiram Eldar, Jul 23 2024

nonn

First differs from A301414 at n = 454: a(454) = 526399743264198303532032000 is not a term of A301414. Is A301414 a subsequence of this sequence? The first 1073 terms of A301414 are in this sequence.
Indices of records of A374902(k)/A374903(k) = A007425(k)/A000005(k).
All the terms are least integers of their prime signature (A025487) since A374902(k)/A374903(k) depends only on the prime signature of k.
The corresponding record values are 1, 3/2, 2, 9/4, 5/2, 3, 15/4, 4, 9/2, 5, ... .

Amiram Eldar, Table of n, a(n) for n = 1..1085

Subsequence of A025487.

Cf. A000005, A002182, A007425, A301414, A308912, A374902, A374903, A374904, A374905, A374906.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; f[p_, e_] := e/2 + 1; f[1] = 1; f[n_] := Times @@ f @@@ FactorInteger[n]; s = {}; fmax = -1; Do[f1 = f[lps[[k]]]; If[f1 > fmax, fmax = f1; AppendTo[s, lps[[k]]]], {k, 1, Length[lps]}]; s

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

Original entry on oeis.org

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]]

cp's OEIS Frontend

A301413 a(n) = A002182(n)/A002110(A108602(n)).

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A059992 Numbers with an increasing number of nonprime divisors.

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A304234 Superior highly composite numbers that are superabundant but not colossally abundant.

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A304235 Colossally abundant numbers that are highly composite, but not superior highly composite.

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A301415 Number of terms m in A002110 such that A301413(k) * A002110(m) is in A002182.

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A301416 Numbers k in A301413 such that k * A002110 (m) is in A002201.

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A340840 Union of the highly composite and superabundant numbers.

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A365900 Highly composite numbers k that remain highly composite when recursively divided by squarefree kernel.

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