This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379753 #12 Jan 03 2025 13:42:14
%S A379753 60,120,240,480,840,1260,1680,2520,3360,5040,7560,10080,15120,20160,
%T A379753 27720,36960,55440,83160,110880,166320,221760,277200,332640,443520,
%U A379753 498960,554400,665280,720720,997920,1081080,1330560,1441440,2162160,2882880,3603600,4324320,5765760
%N A379753 Numbers that set records in A379752.
%C A379753 Proper subset of the intersection of A025487 and A375055.
%C A379753 Conjecture: subset of A332785 = A126706 \ A286708.
%C A379753 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.
%C A379753 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.
%H A379753 Michael De Vlieger, <a href="/A379753/b379753.txt">Table of n, a(n) for n = 1..226</a>
%H A379753 Michael De Vlieger, <a href="/A379753/a379753.txt">Prime Power Decomposition of a(n)</a> for n = 1..226, expanding on the table in the example.
%H A379753 Michael De Vlieger, <a href="/A379753/a379753.png">Plot S(n) = P(omega(n))*m at (x,y) = (m, omega(n))</a>, 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.
%e A379753 Let b(n) = A379752(n).
%e A379753 Table showing exponents of prime power factors of a(n) for n = 1..12.
%e A379753 Example: a(6) = 1260 = 2^2 * 3^2 * 5 * 7, hence we write "2.2.1.1".
%e A379753 n a(n) Exp. b(a(n))
%e A379753 ----------------------------------
%e A379753 1 60 ** 2.1.1 1 6*10
%e A379753 2 120 ** 3.1.1 2 6*20 = 10*12
%e A379753 3 240 * 4.1.1 3 6*40 = 10*24 = 12*20
%e A379753 4 480 5.1.1 4 6*80 = 10*48 = 12*40 = 20*24
%e A379753 5 840 * 3.1.1.1 6 6*140 = 10*84 = 12*70 = 14*60 = 20*42 = 28*30
%e A379753 6 1260 * 2.2.1.1 7
%e A379753 7 1680 * 4.1.1.1 9
%e A379753 8 2520 ** 3.2.1.1 11
%e A379753 9 3360 5.1.1.1 12
%e A379753 10 5040 ** 4.2.1.1 15
%e A379753 11 7560 * 3.3.1.1 16
%e A379753 12 10080 * 5.2.1.1 19
%e A379753 * = a(n) is highly composite (in A002182),
%e A379753 ** = a(n) is superior highly composite (in both A002182 and A002201).
%t A379753 (* Load function f at A025487 *)
%t A379753 r = 0;
%t A379753 s = Select[Union@ Flatten@ f[14][[4 ;; -1]], Not@*SquareFreeQ];
%t A379753 nn = Length[s]; Print[nn];
%t A379753 Reap[Do[k = s[[i]]; If[# > r, r = #; Sow[k]] &@
%t A379753 Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
%t A379753 _?(And[1 < GCD @@ {##},
%t A379753 Nor[Divisible[#2, rad[#1]],
%t A379753 Divisible[#1, rad[#2]] ] ] & @@ # &)], {i, nn}] ][[-1, 1]]
%Y A379753 Cf. A002182, A002201, A025487, A375055, A379752, A379754.
%K A379753 nonn
%O A379753 1,1
%A A379753 _Michael De Vlieger_, Jan 01 2025