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
Keywords
Examples
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.
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.
Programs
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Maple
A253641:=proc(n) if n in {0,1} then 1 else igcd(map(i->i[2], ifactors(n)[2])[]); fi; end: seq(numtheory[sigma](A253641(n)), n=1..120); # Ridouane Oudra, Jun 04 2025
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Mathematica
a[n_] := DivisorSigma[1, GCD @@ FactorInteger[n][[;; , 2]]]; Array[a, 100]
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PARI
a(n) = if (n==1, 1, sigma(gcd(factor(n)[,2]))); \\ Michel Marcus, Jul 16 2021
Formula
If n = Product_{i} p_i^e_i, then a(n) = sigma(gcd()).
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
a(n) = sigma(A253641(n)). - Ridouane Oudra, Jun 04 2025
Comments