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 A376687 #80 Jan 11 2025 03:47:29
%S A376687 24,96,120,240,360,480,840,1080,1680,2160,2520,3360,4320,5040,7560,
%T A376687 10080,15120,27720,30240,55440,60480,83160,110880,151200,166320,
%U A376687 221760,277200,332640,498960,554400,665280,831600,997920,1330560,1441440,1663200,2162160,2882880
%N A376687 Numbers that set records in in A376281.
%C A376687 Proper subset of the intersection A025487 and A379336.
%C A376687 There are three kinds of pairs (d, k/d), d | k, such that gcd(d, k/d) does not equal 1, d, or k/d:
%C A376687 Type A: rad(d) does not divide k/d, and rad(k/d) does not divide d (see A379752), where rad = A007947.
%C A376687 Type B: the squarefree kernel of one divisor divides the other but the reverse is not true (see A379772).
%C A376687 Type C: rad(d) = rad(k/d), i.e., d, k/d, and k are coreful (see A379552).
%C A376687 Conjecture: Numbers k that set records in A376281 do not have type C divisor pairs, i.e., those that are coreful but neither divides the other. This, since type C requires k to be powerful and divisible by cubes of 2 distinct primes (i.e., in A376936). Therefore the record is achieved only through large numbers of type A and B.
%C A376687 Since type A divisor pairs are common for composite k in A375055, this sequence is resembles A379752.
%C A376687 Since d and k/d are both composite, this sequence resembles A059992.
%C A376687 This sequence, to a lesser extent A379752, and a greater extent A059992, contains many highly composite numbers. (See plot of S(n) = union of this sequence and A002182 below, and corresponding graphs in respective other sequences.)
%H A376687 Michael De Vlieger, <a href="/A376687/b376687.txt">Table of n, a(n) for n = 1..241</a>
%H A376687 Michael De Vlieger, <a href="/A376687/a376687.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.
%H A376687 Michael De Vlieger, <a href="/A376687/a376687_1.png">List of (d, k/d)</a>, d < k/d, k = a(n), n = 1..24, such that gcd(d, k/d) is not in {1, d, k/d}, showing type A in light gray, type B in either blue or gold, and type C (if it occurs) in black.
%H A376687 Michael De Vlieger, <a href="/A376687/a376687.txt">Prime power decomposition of a(n)</a>, n = 1..241.
%e A376687 Let b(n) = A376281(n).
%e A376687 Table showing exponents of prime power factors of a(n) for n = 1..20.
%e A376687 Example: a(5) = 360 = 2^3 * 3^2 * 5, hence we write "3.2.1".
%e A376687 n a(n) Exp. b(a(n))
%e A376687 ----------------------------------
%e A376687 1 24 * 3.1 1 4*6
%e A376687 2 96 5.1 2 6*16 = 8*12
%e A376687 3 120 ** 3.1.1 3 4*30 = 6*20 = 10*12
%e A376687 4 240 * 4.1.1 4 6*40 = 8*30 = 10*24 = 12*20
%e A376687 5 360 ** 3.2.1 5 4*90 = 10*36 = 12*30 = 15*24 = 18*20
%e A376687 6 480 5.1.1 6 6*80 = 8*60 = 10*48 = 12*40 = 16*30 = 20*24
%e A376687 7 840 * 3.1.1.1 7
%e A376687 8 1080 3.3.1 9
%e A376687 9 1680 * 4.1.1.1 10
%e A376687 10 2160 4.3.1 11
%e A376687 11 2520 ** 3.2.1.1 13
%e A376687 12 3360 5.1.1.1 14
%e A376687 * = a(n) is highly composite (in A002182),
%e A376687 ** = a(n) is superior highly composite (in both A002182 and A002201).
%t A376687 (* Load function f at A025487 *)
%t A376687 r = 0; s = Select[Union@ Flatten@ f[8][[3 ;; -1]], Not@*SquareFreeQ];
%t A376687 nn = Length[s]; Print[nn]
%t A376687 Reap[Monitor[
%t A376687 Do[k = s[[i]];
%t A376687 If[# > r, r = #; Sow[k]] &@
%t A376687 Count[Transpose@{#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
%t A376687 _?(And[1 < GCD @@ {##}, Mod[#1, #2] != 0,
%t A376687 Mod[#2, #1] != 0] & @@ # &)], {i, nn}], i] ][[-1, 1]]
%Y A376687 Cf. A002182, A002201, A007947, A025487, A059992, A379336, A376281, A377525, A379752.
%K A376687 nonn
%O A376687 1,1
%A A376687 _Michael De Vlieger_, Jan 08 2025