A346403 a(1)=1; for n>1, a(n) gives the sum of the exponents in the different ways to write n as n = x^y, 2 <= x, 1 <= y.

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1

Offset

1,4

Comments

Denoted by tau(n) in Mycielski (1951), Fehér et al. (2006), and Awel and Küçükaslan (2020).

This function depends only on the prime signature of n (see the Formula section).

Links

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

Abdu Awel and M. Küçükaslan, A Note on Statistical Limit and Cluster Points of the Arithmetical Functions a_p(n), gamma(n), and tau(n) in the Sense of Density, Journal of the Indonesian Mathematical Society, Vol. 26, No. 2 (2020), pp. 224-233.

Zoltán Fehér, Béla László, Martin Mačaj and Tibor Šalát, Remarks on arithmetical functions a_p(n), gamma(n), tau(n), Annales Mathematicae et Informaticae, Vol. 33 (2006), pp. 35-43.

Jan Mycielski, Sur les représentations des nombres naturels par des puissances à base et exposant naturels, Colloquium Mathematicum, Vol. 2 (1951), pp. 254-260.

Formula

If n = Product_{i} p_i^e_i, then a(n) = sigma(gcd(<e_i>)).

Sum_{n>=1} (a(n)-1)/n = Pi^2/6 + 1 (= A013661 + 1) (Mycielski, 1951).

Example

4 = 2^2, gcd(2) = 2, sigma(2) = 3, so a(4) = 3. The representations are 4^1 and 2^2, and 1 + 2 = 3.
144 = 2^4 * 3^2, gcd(4,2) = 2, sigma(2) = 3, so a(144) = 3. The representations are 144^1 and 12^2, and 1 + 2 = 3.

Mathematica

a[n_] := DivisorSigma[1, GCD @@ FactorInteger[n][[;; , 2]]]; Array[a, 100]

Prog

(PARI) a(n) = if (n==1, 1, sigma(gcd(factor(n)[, 2]))); \\ Michel Marcus, Jul 16 2021

Crossrefs

Cf. A000203, A013661, A089723.

Sequence in context: A059619 A098950 A318873 * A295924 A357669 A361012

Adjacent sequences: A346400 A346401 A346402 * A346404 A346405 A346406

Keyword

nonn

Author

Amiram Eldar, Jul 15 2021

Status

approved