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
Examples
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.
References
- James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.3.21 on page 100.
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.
Crossrefs
Programs
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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]
Formula
Let f(n) = a(n)/A346010(n) be the sequence of fractions. Then:
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).