A322966 Denominator of Sum_{d | n} 1/rad(d) where rad = A007947.

1, 2, 3, 1, 5, 1, 7, 2, 3, 5, 11, 3, 13, 7, 5, 1, 17, 2, 19, 5, 21, 11, 23, 3, 5, 13, 1, 7, 29, 5, 31, 2, 11, 17, 35, 3, 37, 19, 39, 1, 41, 7, 43, 11, 1, 23, 47, 1, 7, 10, 17, 13, 53, 1, 55, 7, 57, 29, 59, 5, 61, 31, 21, 1, 65, 11, 67, 17, 23, 35, 71, 6, 73, 37, 15, 19, 77, 13, 79, 5

Offset: 1

Author

David S. Metzler, Dec 31 2018

Keywords

• frac
• nonn

Comments

Let rad(n) be the radical of n, which equals the product of all prime factors of n (A007947). Let f(n) = Sum_{d | n} 1/rad(n). The sequence a(n) lists the denominators of the fractions f(n) in lowest terms.

f 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.

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, and a(12) = 3. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3) = 8/3. The denominator of f(12) is 3 hence a(12) = 3.
		

Crossrefs

Cf. A007947 (radical), A322965 (numerators).

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

Programs

Mathematica

Array[Denominator@ DivisorSum[#, 1/Apply[Times, FactorInteger[#][[All, 1]] ] &] &, 80] (* Michael De Vlieger, Jan 19 2019 *)

PARI

a(n) = denominator(sumdiv(n, d, 1/factorback(factor(d)[, 1]))) \\ David A. Corneth, Jan 01 2019