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

%I A367511 #29 Mar 02 2024 13:09:09
%S A367511 1,4,36,48,45360,50400
%N A367511 Highly composite numbers h(k) = A002182(k) such that h >= rad(h)^2, where rad() = A007947().
%C A367511 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.
%C A367511 This sequence is likely finite and full. See Chapter III regarding the structure of "Highly Composite Numbers".
%C A367511 Terms larger than 36 are in A366250; A366250 is in A364702, which is in turn a proper subset of A332785, itself contained in A126706.
%C A367511 36 is in A365308, a proper subset of A303606, contained in A131605, in turn contained in A286708.
%H A367511 Srinivasa Ramanujan, <a href="https://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page9.htm">Highly Composite Numbers</a>, Proc. London Math. Soc. (1916) Vol. 2, No. 14, 347-409.
%H A367511 D. B. Siano and J. D. Siano, <a href="http://wwwhomes.uni-bielefeld.de/achim/julianmanuscript3.pdf">An Algorithm for Generating Highly Composite Numbers</a>, 1994.
%e A367511 Let P(n) = A002110(n).
%e A367511 a(1) = h(1) = 1 since 1 >= 1^2.
%e A367511 a(2) = h(3) = 4 since 4 >= P(1)^2, 4 >= 2^2.
%e A367511 a(3) = h(7) = 36 since 36 >= P(2)^2, 36 >= 6^2.
%e A367511 a(4) = h(8) = 48 since 48 >= P(2)^2, 48 >= 6^2.
%e A367511 a(5) = h(26) = 43560 since 43560 >= P(4)^2, where P(4) = 210, and 210^2 = 44100.
%e A367511 a(6) = h(27) = 50400 since 50400 >= P(4)^2.
%e A367511 Let V(i) = A301414(i) and let P(j) = A002110(j).
%e A367511 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 (*):
%e A367511 V(i)\P(j) 1   2    6   30   210    2310    30030 ...
%e A367511         +---------------------------------------
%e A367511       1 |(1*) 2*   6*
%e A367511       2 |    (4*) 12*  60*
%e A367511       4 |         24* 120*  840*
%e A367511       6 |        (36) 180* 1260*
%e A367511       8 |        (48) 240  1680*
%e A367511      12 |             360  2520   27720*
%e A367511      24 |             720  5040   55440   720720
%e A367511      36 |                  7560   83160  1081080
%e A367511      48 |                 10080  110880  1441440
%e A367511      72 |                 15120  166320  2162160
%e A367511      96 |                 20160  221760  2882880
%e A367511     120 |                 25200  277200  3603600
%e A367511     144 |                        332640  4324320
%e A367511     216 |                (45360) 498960  6486480
%e A367511     240 |                (50400) 554400  7207200
%e A367511     ...
%t A367511 (* First load function f at A025487, then run the following: *)
%t A367511 s = Union@ Flatten@ f[12];
%t A367511 t = Map[DivisorSigma[0, #] &, s];
%t A367511 h = Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]];
%t A367511 Reap[Do[If[# >= Product[Prime[j], {j, PrimeNu[#]}]^2, Sow[#]] &[ h[[i]] ],
%t A367511   {i, Length[h]}] ][[-1, 1]]
%Y A367511 Cf. A001221, A002110, A002182, A007947, A025487, A108602, A126706, A131605, A168263, A286708, A301413, A301414, A303606, A332785, A365308, A362702, A366250.
%K A367511 nonn,more
%O A367511 1,2
%A A367511 _Michael De Vlieger_, Feb 08 2024