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 A322447 #19 Feb 01 2024 09:41:33
%S A322447 1,2,4,8,12,16,24,32,48,72,96,144,192,288,384,576,864,1152,1728,2304,
%T A322447 3456,4608,5184,6912,10368,13824,20736,27648,41472,55296,62208,82944,
%U A322447 124416,165888,207360,248832,331776,373248,414720,497664,622080,746496,829440,995328
%N A322447 Numbers k where Sum_{d | k} 1/rad(d) increases to a record.
%C A322447 Let rad(n) be the radical of n, which equals the product of all prime factors of n (A007947). Let g(n) = 1/rad(n) and let f(n) = Sum_{d | n} g(d). This is a multiplicative function whose value on a prime power is f(p^k) = 1 + k/p. Hence f is a weighted divisor-counting function that weights divisors d higher when they have few and small prime divisors themselves. This sequence lists the values where f(n) increases to a record, analogously to highly composite numbers (A002182) or superabundant numbers (A004394). The numbers in this sequence are much smoother than those in the other two sequences, since the definition of f(n) strongly disfavors a lack of smoothness in n.
%H A322447 Charlie Neder, <a href="/A322447/b322447.txt">Table of n, a(n) for n = 1..200</a>
%e A322447 The divisors of 12 are 1,2,3,4,6,12, so f(12) = 1 + (1/2) + (1/3) + (1/2) + (1/6) + (1/6) = 8/3, which exceeds f(n) for n = 1,...,11. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3).
%e A322447 f(207360) = f(2^9)*f(3^4)*f(5) = (11/2)*(7/3)*(6/5) = 15.4, which exceeds f(n) for n < 207360. (Note that this is the first value of the sequence that is divisible by 5; earlier values are all 3-smooth.)
%t A322447 rad[n_] := Times @@ (First@# & /@ FactorInteger@n); f[n_] := DivisorSum[n, 1/rad[#] &]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^6}]; s (* _Amiram Eldar_, Dec 08 2018 *)
%o A322447 (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
%o A322447 lista(nn) = {my(m=0, newm); for (n=1, nn, newm = sumdiv(n, d, 1/rad(d)); if (newm > m, m = newm; print1(n, ", ")););} \\ _Michel Marcus_, Dec 09 2018
%Y A322447 Cf. A007947 (radical), A002182, A004394.
%Y A322447 Also smooth numbers: A003586, A051037, A002473.
%K A322447 nonn
%O A322447 1,2
%A A322447 _David S. Metzler_, Dec 08 2018