OFFSET
1,1
COMMENTS
This sequence seems to be rich in highly composite numbers, the prime shape of a(n) resembles that of highly composite numbers, with long tails of large prime factors with multiplicity 1.
Terms not in A002182 are not all of the form 2^5 * prime(i..j), 1 < i < j, for example, a(24) = 443520 = 2^7 * 3^2 * 5 * 7 * 11.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..226
Michael De Vlieger, Prime Power Decomposition of a(n) for n = 1..226, expanding on the table in the example.
Michael De Vlieger, Plot S(n) = P(omega(n))*m at (x,y) = (m, omega(n)), where S is the union of A002182 and this sequence, P is A002110, omega is A001221, and only select m that harbor S(n) shown. Shows the coincidence of many terms in this sequence with A002182. Blue represents m in A002182, gold m in both A002182 and this sequence; dark blue represents m in A002201 (and also in A002182), orange m in both A002201 and this sequence; red indicates terms in this sequence that are not in A002182. Green highlights terms in A002182 but are not determined to be in this sequence.
EXAMPLE
Let b(n) = A379752(n).
Table showing exponents of prime power factors of a(n) for n = 1..12.
Example: a(6) = 1260 = 2^2 * 3^2 * 5 * 7, hence we write "2.2.1.1".
n a(n) Exp. b(a(n))
----------------------------------
1 60 ** 2.1.1 1 6*10
2 120 ** 3.1.1 2 6*20 = 10*12
3 240 * 4.1.1 3 6*40 = 10*24 = 12*20
4 480 5.1.1 4 6*80 = 10*48 = 12*40 = 20*24
5 840 * 3.1.1.1 6 6*140 = 10*84 = 12*70 = 14*60 = 20*42 = 28*30
6 1260 * 2.2.1.1 7
7 1680 * 4.1.1.1 9
8 2520 ** 3.2.1.1 11
9 3360 5.1.1.1 12
10 5040 ** 4.2.1.1 15
11 7560 * 3.3.1.1 16
12 10080 * 5.2.1.1 19
* = a(n) is highly composite (in A002182),
MATHEMATICA
(* Load function f at A025487 *)
r = 0;
s = Select[Union@ Flatten@ f[14][[4 ;; -1]], Not@*SquareFreeQ];
nn = Length[s]; Print[nn];
Reap[Do[k = s[[i]]; If[# > r, r = #; Sow[k]] &@
Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
_?(And[1 < GCD @@ {##},
Nor[Divisible[#2, rad[#1]],
Divisible[#1, rad[#2]] ] ] & @@ # &)], {i, nn}] ][[-1, 1]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Jan 01 2025
STATUS
approved