A301413 -id:A301413 - cp's OEIS frontend

A301414 Distinct terms of A301413 in ascending order: terms k in A301413 that have at least one number m such that k * A002110(m) is a highly composite number (A002182) with m distinct prime factors.

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

Offset: 1

Michael De Vlieger, Apr 09 2018

nonn

Given that highly composite numbers (HCNs) are products of primorials, we note the following:
1. The only odd term is 1.
2. The only primorials, i.e., terms in A002110, are {1, 2, 6}, consequently the only squares in A002182 are {1, 4, 36}.
3. The only terms in A000079 are {1, 2, 4, 8}. These produce {1, 2, 6}, {4, 12, 30}, {24, 120, 840}, and {48, 240, 1680}, in A002182 respectively.
4. This sequence is a subset of A025487, which is a subset of A055932.
Also given that A002182 strictly increases, we note that i <= m <= j, integers, for which P = k * A002110(m) produces HCNs. As we increment m we increase the rank of the tensor of prime divisor power ranges and double the number of divisors. However, we may have another term P' = a * A002110(b) for a > k and b < (j + 1) such that P' < P yet tau(P') >= tau(P). This P' is in A002182 and has increased tau by the lengthening of the power ranges for relatively small primes via some composite b instead of increasing the rank of the tensor. Since A002182 strictly increases, we have a limited range for m.
There are 19 terms also in A002182: 1, 2, 4, 6, 12, 24, 36, 48, 120, 240, 360, 720, 5040, 7560, 10080, 15120, 20160, 50400, 17297280.
Let n = A002110(m), and consider the ordered pair (n, k). In a plot of ordered pairs that produce m in A002182, we have the first terms of A002182 thus: (0,1), (1,1), (1,2), (2,1), (2,2), (2,4), (2,6), (2,8), (3,2), (3,4), (3,6), (3,8), (3,12), etc.

			Plot of (n,k) with n in A002110 and k a term in this sequence such that A002110(n) * k is in A002182. Asterisks denote products that are in A002201.
   {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*
                           {4,36}   {5,36}    {6,36}
                             7560    83160   1081080
                           {4,48}   {5,48}    {6,48}
                            10080   110880   1441440*
                            ...     ...      ...       ...

Michael De Vlieger, Table of n, a(n) for n = 1..8000
Michael De Vlieger, On a graph of highly composite numbers
Michael De Vlieger, Plot of 779674 HCNs as A002110(y) * A301414(x) at (x,y).
A. Flammenkamp, Highly composite numbers

Cf. A000079, A002110, A002182, A002201, A025487, A055932, A275055.

(* First load b-file from A002182 minus any comments therein *)
s = Import["b002182.txt","Data"][[All,-1]];
(* Alternatively, download Flammenkamp dataset, decompress and rename to "HCN.txt", then decode using the following in place of s above *)
s = Times @@ {Times @@ Prime@ Range@ ToExpression@ First@ #1, If[# == {}, 1, Times @@ MapIndexed[Prime[First@ #2]^#1 &, #]] &@ DeleteCases[-1 + Flatten@ Map[If[StringFreeQ[#, "^"], ToExpression@ #, ConstantArray[#1, #2] & @@ ToExpression@ StringSplit[#, "^"]] &, #2], 0]} & @@ TakeDrop[Drop[StringSplit@ #, 2], 1] & /@ Import["HCN.txt", "Data"];
Union@ Array[#1/Product[Prime@ i, {i, #2}] & @@ {#, PrimeNu@ #} &@ s[[#]] &, Length@ s]

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]

A305056 a(n) = A004394(n)/A002110(A001221(A004394(n))).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 6, 8, 2, 4, 6, 8, 12, 24, 4, 6, 8, 12, 24, 48, 72, 120, 12, 24, 48, 72, 120, 144, 240, 288, 24, 48, 72, 120, 144, 240, 288, 360, 720, 72, 120, 144, 240, 288, 360, 720, 72, 1440, 2160, 120, 144, 240, 288, 360, 720, 1440, 2160, 2880, 4320, 5040

Offset: 1

Michael De Vlieger, Jul 01 2018

nonn

This sequence is analogous to A301413, which pertains to A002182.
Since A002182(20) = 7560 is not in A004394, a(20) =/= A301413(20), i.e., the former is 36, the latter 48. (The number 36 is not in this sequence, but is in A301413.)
A004394(50) = 120*A002110(8) is the smallest number in A004394 but not in A002182; in A004394 we have 120*A002110(m) for 4 <= m <= 8, while in A002110 we have 120*A002110(m) for 4 <= m <= 7. Therefore this sequence has one more instance of 120 (= a(50)) than exists in A301413.

			Let m be a value in this sequence. The table below shows m*A002110(A001221(A004394(k))). Columns are A001221(A004394(k)), rows are m whose products m*A002110(A001221(A004394(k))) appear in A004394 are in this sequence. Numbers in A004394 that also appear in A004490 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*
Up to this point, the graph of this sequence and that of A301413 are identical; the asterisks pertain to numbers in A002201 in the case of A301413, but all the numbers on the graph are found in both A004490 and A002201, i.e., in A224078. The next two rows of the graph of A301413:
       0     1     2      3      4       5         6  ...
      +----------------------------------------------------
  36  |                         7560   83160   1081080
  48  |                        10080  110880   1441440*
  ...
but this sequence does not have row m = 36, as {7560, 83160, 1081080} are not in A004394.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A001221, A002110, A004394, A301413.

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

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]]

A365901 Irregular triangle read by rows giving trajectory beginning with A002182(n) under recursion of x -> f(x) until reaching 1, where f(x) = x/rad(x), rad(x) = A007947(x).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 1, 12, 2, 1, 24, 4, 2, 1, 36, 6, 1, 48, 8, 4, 2, 1, 60, 2, 1, 120, 4, 2, 1, 180, 6, 1, 240, 8, 4, 2, 1, 360, 12, 2, 1, 720, 24, 4, 2, 1, 840, 4, 2, 1, 1260, 6, 1, 1680, 8, 4, 2, 1, 2520, 12, 2, 1, 5040, 24, 4, 2, 1, 7560, 36, 6, 1

Offset: 1

Michael De Vlieger, Oct 11 2023

Let h(n) = A002182(n).

Since highly composite numbers h(n) are products of primorials (i.e., in A025487), the squarefree kernel is always a primorial (i.e., in A002110), and the trajectory always reaches 1.

			Row 1 = {1} since h(1) = 1, already 1.
Row 2 = {2, 1} since h(2) = 2, 2/rad(2) = 2/2 = 1, reaching 1.
Row 3 = {4, 2, 1} since h(3) = 4, 4/rad(4) = 4/2 = 2, and we follow the trajectory of 2 thereafter.
Row 6 = {24, 4, 2, 1} since h(6) = 24, 24/rad(24) = 24/6 = 4, and we follow the trajectory of 4 thereafter.
Row 14 = {720, 24, 4, 2, 1} since h(14) = 720, 720/rad(720) = 720/30 = 24, which appends row 6 thereafter.
.
First rows of this sequence:
row 1:   1
    2:   2, 1
    3:   4, 2, 1
    4:   6, 1
    5:  12, 2, 1
    6:  24, 4, 2, 1
    7:  36, 6, 1
    8:  48, 8, 4, 2, 1
    9:  60, 2, 1
   10: 120, 4, 2, 1
   11: 180, 6, 1
   12: 240, 8, 4, 2, 1
   ...

Michael De Vlieger, Table of n, a(n) for n = 1..11397 (rows n = 1..1200, flattened)
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..1053678, showing 2^16 rows.

Cf. A002110, A002182, A007947, A112779, A301413, A365900.

a2182 = Import["https://oeis.org/A002182/b002182.txt", "Data"][[All, -1]]; Array[NestWhileList[#/(Times @@ FactorInteger[#][[All, 1]]) &, a2182[[#]], # > 1 &] &, 20] // TableForm

Length of row n is A112779(n)+1.

T(n,2) = A301413(n), n > 1.

A367511 Highly composite numbers h(k) = A002182(k) such that h >= rad(h)^2, where rad() = A007947().

Original entry on oeis.org

1, 4, 36, 48, 45360, 50400

Offset: 1

Michael De Vlieger, Feb 08 2024

Alternatively, this sequence lists h(k) such that A301413(k) >= A002110(A108602(k)), where A301413 is the "variable part" v described on page 5 of 12 of the Siano paper.
This sequence is likely finite and full. See Chapter III regarding the structure of "Highly Composite Numbers".
Terms larger than 36 are in A366250; A366250 is in A364702, which is in turn a proper subset of A332785, itself contained in A126706.
36 is in A365308, a proper subset of A303606, contained in A131605, in turn contained in A286708.

			Let P(n) = A002110(n).
a(1) = h(1) = 1 since 1 >= 1^2.
a(2) = h(3) = 4 since 4 >= P(1)^2, 4 >= 2^2.
a(3) = h(7) = 36 since 36 >= P(2)^2, 36 >= 6^2.
a(4) = h(8) = 48 since 48 >= P(2)^2, 48 >= 6^2.
a(5) = h(26) = 43560 since 43560 >= P(4)^2, where P(4) = 210, and 210^2 = 44100.
a(6) = h(27) = 50400 since 50400 >= P(4)^2.
Let V(i) = A301414(i) and let P(j) = A002110(j).
Plot of highly composite h = V(i)*P(j) at (x,y) = (j,i), i = 1..16, j = 1..7, showing h in this sequence in parentheses, and h in A168263 marked with an asterisk (*):
V(i)\P(j) 1   2    6   30   210    2310    30030 ...
        +---------------------------------------
      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
     36 |                  7560   83160  1081080
     48 |                 10080  110880  1441440
     72 |                 15120  166320  2162160
     96 |                 20160  221760  2882880
    120 |                 25200  277200  3603600
    144 |                        332640  4324320
    216 |                (45360) 498960  6486480
    240 |                (50400) 554400  7207200
    ...

Srinivasa Ramanujan, Highly Composite Numbers, Proc. London Math. Soc. (1916) Vol. 2, No. 14, 347-409.
D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite Numbers, 1994.

Cf. A001221, A002110, A002182, A007947, A025487, A108602, A126706, A131605, A168263, A286708, A301413, A301414, A303606, A332785, A365308, A362702, A366250.

(* First load function f at A025487, then run the following: *)
s = Union@ Flatten@ f[12];
t = Map[DivisorSigma[0, #] &, s];
h = Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]];
Reap[Do[If[# >= Product[Prime[j], {j, PrimeNu[#]}]^2, Sow[#]] &[ h[[i]] ],
  {i, Length[h]}] ][[-1, 1]]

cp's OEIS Frontend

A301414 Distinct terms of A301413 in ascending order: terms k in A301413 that have at least one number m such that k * A002110(m) is a highly composite number (A002182) with m distinct prime factors.

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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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A305056 a(n) = A004394(n)/A002110(A001221(A004394(n))).

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

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A365901 Irregular triangle read by rows giving trajectory beginning with A002182(n) under recursion of x -> f(x) until reaching 1, where f(x) = x/rad(x), rad(x) = A007947(x).

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Formula

A367511 Highly composite numbers h(k) = A002182(k) such that h >= rad(h)^2, where rad() = A007947().

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