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

1, 4, 36, 48, 45360, 50400

Offset: 1

Author

Michael De Vlieger, Feb 08 2024

Keywords

• nonn
• more

Comments

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.

Examples

			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
    ...
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

(* 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]]