A322965 - OEIS

%I #30 Dec 09 2023 07:09:58

%S 1,3,4,2,6,2,8,5,5,9,12,8,14,12,8,3,18,5,20,12,32,18,24,10,7,21,2,16,

%T 30,12,32,7,16,27,48,10,38,30,56,3,42,16,44,24,2,36,48,4,9,21,24,28,

%U 54,3,72,20,80,45,60,16,62,48,40,4,84,24,68,36,32,72,72,25,74,57,28

%N Numerator of Sum_{d | n} 1/rad(d).

%C 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. The sequence a(n) lists the numerators of the fractions f(n) in lowest terms.

%C If p is prime, then a(p^k) = p+k if p does not divide k, 1 + k/p if it does. In particular, a(p^p) = 2. - _Robert Israel_, Jan 25 2019

%H Robert Israel, <a href="/A322965/b322965.txt">Table of n, a(n) for n = 1..10000</a>

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A322966(k) = zeta(2)*zeta(3)/zeta(6) (A082695). - _Amiram Eldar_, Dec 09 2023

%e 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 and a(12) = 8. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3) = 8/3.

%p rad:= n -> convert(numtheory:-factorset(n),`*`):

%p f:= proc(n) numer(add(1/rad(d),d=numtheory:-divisors(n))) end proc:

%p map(f, [$1..100]); # _Robert Israel_, Jan 25 2019

%t Array[Numerator@ DivisorSum[#, 1/Apply[Times, FactorInteger[#][[All, 1]]] &] &, 71] (* _Michael De Vlieger_, Jan 19 2019 *)

%o (PARI) rad(n) = factorback(factor(n)[, 1]); \\ A007947

%o a(n) = numerator(sumdiv(n, d, 1/rad(d))); \\ _Michel Marcus_, Jan 10 2019

%Y Cf. A007947 (radical), A322966 (denominators), A008473 (unreduced numerators, i.e., f(n)*rad(n)), A082695.

%Y Numbers n where f(n) increases to a record: A322447.

%K frac,nonn

%O 1,2

%A _David S. Metzler_, Dec 31 2018

%E More terms from _Michel Marcus_, Jan 19 2019