A383159 The sum of the maximum exponents in the prime factorizations of the unitary divisors of n.

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 6, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 11, 1, 3, 5, 6, 3, 7, 1, 5, 3, 7, 1, 8, 1, 3, 5, 5, 3, 7, 1, 9, 4, 3, 1, 11, 3, 3, 3

Offset

1,4

Comments

First differs from A032741 at n = 36, and from A305611 and A325770 at n = 30.

a(n) depends only on the prime signature of n (A118914).

Links

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

Amiram Eldar, Plot of abs((Sum_{k=1..n} a(k))/(c_1*n*log(n) + c_2*n) - 1) for n = 10^(1..10).

Formula

a(n) = Sum_{d|n, gcd(d, n/d) = 1} A051903(d).

a(n) = A034444(n) * A383160(n)/A383161(n).

a(n) <= A383156(n), with equality if and only if n is squarefree (A005117).

a(n) = utau(n, 2) - 1 + Sum_{k=2..A051903(n)} k * (utau(n, k+1) - utau(n, k)), where utau(n, k) is the number of k-free unitary divisors of n (k-free numbers are numbers that are not divisible by a k-th power other than 1). For a given k >= 2, utau(n, k) is a multiplicative function with utau(p^e, k) = 2 if e < k, and 1 otherwise. E.g., utau(n, 2) = A056671(n), utau(n, 3) = A365498(n), and utau(n, 4) = A365499(n).

Sum_{k=1..n} a(k) ~ c_1 * n * log(n) + c_2 * n, where c_1 = c(2) + Sum_{k>=3} (k-1) * (c(k) - c(k-1)) = 0.91974850283445458744..., c(k) = Product_{p prime} (1 - 1/p^2 - 1/p^k + 1/p^(k+1)), c_2 = -1 + (2*gamma - 1)*c_1 + d(2) + Sum_{k>=3} (k-1) * (d(k) - d(k-1)) = -0.50780794945146599739..., d(k) = c(k) * Sum_{p prime} (2*p^(k-1) + k*p - k - 1) * log(p) / (p^(k+1) - p^(k-1) - p + 1), and gamma is Euler's constant (A001620).

Example

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) = 0 + 2 = 2.
12 has 4 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) = 0 + 1 + 2 + 2 = 5.

Mathematica

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := DivisorSum[n, emax[#] &, CoprimeQ[#, n/#] &]; Array[a, 100]
(* second program: *)
a[n_] := If[n == 1, 0, Module[{e = FactorInteger[n][[;; , 2]], emax, v}, emax = Max[e]; v = Table[Times @@ (If[# < k + 1, 2, 1] & /@ e), {k, 1, emax}]; v[[1]] + Sum[k*(v[[k]] - v[[k - 1]]), {k, 2, emax}] - 1]]; Array[a, 100]

Prog

(PARI) emax(n) = if(n == 1, 0, vecmax(factor(n)[, 2]));
a(n) = sumdiv(n, d, (gcd(d, n/d) == 1) * emax(d));
(PARI) a(n) = if(n == 1, 0, my(e = factor(n)[, 2], emax = vecmax(e), v); v = vector(emax, k, vecprod(apply(x ->if(x < k+1, 2, 1), e))); v[1] + sum(k = 2, emax, k * (v[k]-v[k-1])) - 1);

Crossrefs

The unitary version of A383156.

Cf. A005117, A034444, A051903, A056671, A077610, A365498, A365499, A383160, A383161.

Cf. A032741, A305611, A325770.

Sequence in context: A294928 A325770 A305611 * A325765 A032741 A364818

Adjacent sequences: A383156 A383157 A383158 * A383160 A383161 A383162

Keyword

nonn,easy

Author

Amiram Eldar, Apr 18 2025

Status

approved