A301414 -id:A301414 - cp's OEIS frontend

A365902 Irregular triangle of highly composite numbers h(n) = A002182(n) arranged first according to rad(h(n))/h(n) then by rad(h(n)), where rad(n) = A007947(n).

1, 2, 6, 4, 12, 60, 24, 120, 840, 36, 180, 1260, 48, 240, 1680, 360, 2520, 27720, 720, 5040, 55440, 720720, 7560, 83160, 1081080, 10080, 110880, 1441440, 15120, 166320, 2162160, 36756720, 698377680, 20160, 221760, 2882880, 25200, 277200, 3603600, 61261200, 332640

Offset: 1

Michael De Vlieger, Oct 12 2023

rad(h(n)) = P(omega(h(n))), where P(n) = A002110(n) and omega(n) = A001221(n).

This sequence merely lists terms in row n, it does not reflect S(n,j) = A301414(n)*P(j), where P(j) = rad(A301414(n)*P(j)), since least j > 0 for n > 1.

			Row 1 contains the products of A301414(1) = 1 and each of P(0) = 1, P(1) = 2, and P(2) = 6.
Row 2 contains the products of A301414(2) = 2 and each of P(1), P(2), and P(3) = 30.
Row 3 contains the products of A301414(3) = 4 and each of P(2) and P(3), etc.
Table of first rows of S(n,j), where for S(n,j) = A002182(i), j = A108602(i):
  n\j | 0  1   2    3     4       5
  ----------------------------------
    1 | 1, 2,  6
    2 |    4, 12,  60
    3 |       24, 120
    4 |       36, 180, 1260
    5 |       48, 240, 1680
    6 |           360, 2520,  27720
    7 |           720, 5040, 720720, etc.
In this sequence T(n,k) we have the following:
1: 1, 2, 6;
2: 4, 12, 60;
3: 24, 120;
4: 36, 180, 1260;
5: 48, 240, 1680;
6: 360, 2520, 27720;
7: 720, 5040, 720720; etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10598 (rows n = 1..640, flattened)
Michael De Vlieger, Logarithmic scatterplot of log_10 a(n), n = 1..10^4.
Michael De Vlieger, Image showing 8000 rows of this sequence (715118 HCNs), representing each term with a black pixel, extracted from the Flammenkamp dataset of 779674 HCNs. Note, the HCNs shown are not the smallest; some HCNs are clipped away from the bottom of this image where uncertainty in the completion of rows is expected.

Cf. A001221, A002110, A002182, A007947, A108602, A301414, A301415.

nn = 8; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
MapIndexed[Set[P[First[#2]], #1] &, FoldList[Times, Prime@ Range[nn + 1]]];
a2182 = Import["https://oeis.org/A002182/b002182.txt", "Data"][[All, -1]];
TakeWhile[
   SplitBy[SortBy[
     Map[{#1/#2, PrimeNu[#2], #1} & @@ {#, rad[#]} &,
      TakeWhile[a2182, rad[#] <= P[nn] &]], #[[1 ;; 2]] &,
     LexicographicOrder], First],
   FreeQ[a2182, #1 P[#2 + 1]] & @@ #[[-1, 1 ;; 2]] &][[All, All, -1]] // Flatten

Let i = least j such that A301414(n)*A002110(j) is in A002182.
This sequence is T(n,k) = S(n,j-i+1).
Length of row n = A301415(n).

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

A365902 Irregular triangle of highly composite numbers h(n) = A002182(n) arranged first according to rad(h(n))/h(n) then by rad(h(n)), where rad(n) = A007947(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula