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A346009
a(n) is the numerator of the average number of distinct prime factors of the divisors of n.
9
0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 7, 1, 1, 1, 4, 1, 7, 1, 7, 1, 1, 1, 5, 2, 1, 3, 7, 1, 3, 1, 5, 1, 1, 1, 4, 1, 1, 1, 5, 1, 3, 1, 7, 7, 1, 1, 13, 2, 7, 1, 7, 1, 5, 1, 5, 1, 1, 1, 5, 1, 1, 7, 6, 1, 3, 1, 7, 1, 3, 1, 17, 1, 1, 7, 7, 1, 3, 1, 13, 4, 1, 1, 5, 1, 1
OFFSET
1,4
LINKS
R. L. Duncan, Note on the divisors of a number, The American Mathematical Monthly, Vol. 68, No. 4 (1961), pp. 356-359.
Sébastien Gaboury, Sur les convolutions de fonctions arithmétiques, M.Sc. thesis, Laval University, Quebec, 2007.
FORMULA
Let f(n) = a(n)/A346010(n) be the sequence of fractions. Then:
f(n) = A062799(n)/A000005(n).
f(n) = (Sum_{p prime, p|n} d(n/p))/d(n), where d(n) is the number of divisors of n (A000005).
f(n) depends only on the prime signature of n: If n = Product_{i} p_i^e_i, then a(n) = Sum_{i} e_i/(e_i + 1).
f(p) = 1/2 for prime p.
f(n) = 1 for squarefree semiprimes n (A006881).
Sum_{k=1..n} f(k) ~ (1/2) * A013939(n) + C*n + O(n/log(n)) ~ n*log(log(n))/2 + (B/2 + C)*n + O(n/log(n)), where B is Mertens's constant (A077761) and C = A346011 (Duncan, 1961).
EXAMPLE
The fractions begin with 0, 1/2, 1/2, 2/3, 1/2, 1, 1/2, 3/4, 2/3, 1, 1/2, 7/6, ...
f(2) = 1/2 since 2 has 2 divisors, 1 and 2, and (omega(1) + omega(2))/2 = (0 + 1)/2 = 1/2.
f(6) = 1 since 6 has 4 divisors, 1, 2, 3 and 6 and (omega(1) + omega(2) + omega(3) + omega(6))/4 = (0 + 1 + 1 + 2)/4 = 1.
MATHEMATICA
a[n_] := Numerator[DivisorSum[n, PrimeNu[#] &]/DivisorSigma[0, n]]; Array[a, 100]
(* or *)
f[p_, e_] := e/(e+1); a[1] = 0; a[n_] := Numerator[Plus @@ f @@@ FactorInteger[n]]; Array[a, 100]
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Amiram Eldar, Jul 01 2021
STATUS
approved