A346009 a(n) is the numerator of the average number of distinct prime factors of the divisors of n.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A000005, A001221, A006881, A013939, A062799, A077761, A346010 (denominators), A346011.

Sequence in context: A373378 A337619 A183214 * A344756 A101221 A086291

Adjacent sequences: A346006 A346007 A346008 * A346010 A346011 A346012

Keyword

nonn,frac

Author

Amiram Eldar, Jul 01 2021

Status

approved