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).
a(n) = sigma(A052409(n)), for n>1. - Ridouane Oudra, Nov 23 2024
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.
MAPLE
A052409:=n->igcd(map(i->i[2], ifactors(n)[2])[]):
1, seq(numtheory[sigma](A052409(n)), n=2..120); # Ridouane Oudra, Nov 23 2024
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 15 2021
STATUS
approved