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

1,2

Comments

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.

Links

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.

Formula

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).

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A163755 A100851 A351500 * A303823 A286712 A268716

Adjacent sequences: A365899 A365900 A365901 * A365903 A365904 A365905

Keyword

nonn,tabf

Author

Michael De Vlieger, Oct 12 2023

Status

approved