A322447 Numbers k where Sum_{d | k} 1/rad(d) increases to a record.

1, 2, 4, 8, 12, 16, 24, 32, 48, 72, 96, 144, 192, 288, 384, 576, 864, 1152, 1728, 2304, 3456, 4608, 5184, 6912, 10368, 13824, 20736, 27648, 41472, 55296, 62208, 82944, 124416, 165888, 207360, 248832, 331776, 373248, 414720, 497664, 622080, 746496, 829440, 995328

Offset: 1

Author

David S. Metzler, Dec 08 2018

Keywords

• nonn

Comments

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.

Examples

			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).
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.)
		

Links

• Charlie Neder, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A007947 (radical), A002182, A004394.

Also smooth numbers: A003586, A051037, A002473.

Programs

Mathematica

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 *)

PARI

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
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