A383160 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the unitary divisors of n.
0, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 2, 1, 5, 1, 5, 3, 3, 1, 7, 1, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 3, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 1, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 11, 1, 3, 5, 3, 3, 7, 1, 5, 3, 7, 1, 2, 1, 3, 5, 5, 3, 7, 1, 9, 2, 3, 1, 11, 3, 3, 3
Offset: 1
Examples
Fractions begin with 0, 1/2, 1/2, 1, 1/2, 3/4, 1/2, 3/2, 1, 3/4, 1/2, 5/4, ... 4 has 2 unitary divisors: 1 and 4 = 2^2. The maximum exponents in their prime factorizations are 0 and 2, respectively. Therefore, a(4) = numerator((0 + 2)/2) = numerator(1) = 1. 12 has 4 unitary divisors: 1, 3 = 3^1, 4 = 2^2 and 12 = 2^2 * 3. The maximum exponents in their prime factorizations are 0, 1, 2 and 2, respectively. Therefore, a(12) = numerator((0 + 1 + 2 + 2)/4) = numerator(5/4) = 5.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Amiram Eldar, Plot of (Sum_{k=1..n} a(k)/A383161(k))/(c_1*n - c_2*n/sqrt(log(n))) for n = 10^(1..10).
Crossrefs
Programs
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Mathematica
emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Numerator[DivisorSum[n, emax[#] &, CoprimeQ[#, n/#] &] / 2^PrimeNu[n]]; Array[a, 100]
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PARI
emax(n) = if(n == 1, 0, vecmax(factor(n)[,2])); a(n) = my(f = factor(n)); numerator(sumdiv(n, d, emax(d) * (gcd(d, n/d) == 1)) / (1 << omega(f)));
Formula
a(n) = numerator(Sum_{d|n, gcd(d, n/d) = 1} A051903(d) / A034444(n)) = numerator(A383159(n) / A034444(n)).
a(n)/A383161(n) >= A383157(n)/A383158(n), with equality if and only if n is either squarefree (A005117) or a power of prime (A000961), i.e., n is not in A126706.
a(n)/A383161(n) < 1 if and only if n is squarefree.
Sum_{k=1..n} a(k)/A383161(k) ~ c_1 * n - c_2 * n / sqrt(log(n)), where c = m(2) + Sum_{k>=3} (k-1) * (m(k) - m(k-1)) = 1.36914082067166047512... is the asymptotic mean of the fractions a(k)/A383161(k), m(k) = Product_{p prime} (1 - 1/(2*p^k)) is the asymptotic mean of the ratio between the number of k-free unitary divisors and the number of unitary divisors. e.g., m(2) = A383057 and m(3) = A383058, and c_2 = A345288/sqrt(Pi) = 0.6189064491320478904... .
Comments