OFFSET
1,2
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".
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.
EXAMPLE
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.
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
...
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]]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Michael De Vlieger, Feb 08 2024
STATUS
approved