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%I #19 Nov 23 2024 08:31:52
%S 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,
%T 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,
%U 1,1,1,1,1,1,1,1,1,1,1,1,7,1,1,1,1,1,1
%N 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.
%C Denoted by tau(n) in Mycielski (1951), Fehér et al. (2006), and Awel and Küçükaslan (2020).
%C This function depends only on the prime signature of n (see the Formula section).
%H Amiram Eldar, <a href="/A346403/b346403.txt">Table of n, a(n) for n = 1..10000</a>
%H Abdu Awel and M. Küçükaslan, <a href="http://dx.doi.org/10.22342/jims.26.2.808.224-233">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</a>, Journal of the Indonesian Mathematical Society, Vol. 26, No. 2 (2020), pp. 224-233.
%H Zoltán Fehér, Béla László, Martin Mačaj and Tibor Šalát, <a href="https://eudml.org/doc/128780">Remarks on arithmetical functions a_p(n), gamma(n), tau(n)</a>, Annales Mathematicae et Informaticae, Vol. 33 (2006), pp. 35-43.
%H Jan Mycielski, <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm2/cm2140.pdf">Sur les représentations des nombres naturels par des puissances à base et exposant naturels</a>, Colloquium Mathematicum, Vol. 2 (1951), pp. 254-260.
%F If n = Product_{i} p_i^e_i, then a(n) = sigma(gcd(<e_i>)).
%F Sum_{n>=1} (a(n)-1)/n = Pi^2/6 + 1 (= A013661 + 1) (Mycielski, 1951).
%F a(n) = sigma(A052409(n)), for n>1. - _Ridouane Oudra_, Nov 23 2024
%e 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.
%e 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.
%p A052409:=n->igcd(map(i->i[2], ifactors(n)[2])[]):
%p 1, seq(numtheory[sigma](A052409(n)), n=2..120); # _Ridouane Oudra_, Nov 23 2024
%t a[n_] := DivisorSigma[1, GCD @@ FactorInteger[n][[;; , 2]]]; Array[a, 100]
%o (PARI) a(n) = if (n==1, 1, sigma(gcd(factor(n)[,2]))); \\ _Michel Marcus_, Jul 16 2021
%Y Cf. A000203, A013661, A089723, A052409.
%K nonn
%O 1,4
%A _Amiram Eldar_, Jul 15 2021