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A322447 Numbers k where Sum_{d | k} 1/rad(d) increases to a record. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

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

EXAMPLE

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.)

MAPLE

with(numtheory): P:=proc(q) local a, b, k, n, x;

b:=[]; x:=0; for n from 1 to q do

a:=1+add(1/convert(factorset(k), `*`), k=divisors(n) minus {1});

if a>x then x:=a; b:=[op(b), n]; fi; od; op(b); end: P(10^6);

# Paolo P. Lava, Jan 11 2019

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 *)

PROG

(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

CROSSREFS

Cf. A007947 (radical), A002182, A004394.

Also smooth numbers: A003586, A051037, A002473.

Sequence in context: A336496 A317804 A328524 * A170892 A246468 A326807

Adjacent sequences:  A322444 A322445 A322446 * A322448 A322449 A322450

KEYWORD

nonn

AUTHOR

David S. Metzler, Dec 08 2018

STATUS

approved

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Last modified May 18 01:26 EDT 2021. Contains 343992 sequences. (Running on oeis4.)