A254926 There are a(n) numbers m such that 1 <= m <= n and gcd(m,n) is cubefree.
1, 2, 3, 4, 5, 6, 7, 7, 9, 10, 11, 12, 13, 14, 15, 14, 17, 18, 19, 20, 21, 22, 23, 21, 25, 26, 26, 28, 29, 30, 31, 28, 33, 34, 35, 36, 37, 38, 39, 35, 41, 42, 43, 44, 45, 46, 47, 42, 49, 50, 51, 52, 53, 52, 55, 49, 57, 58, 59, 60, 61, 62, 63, 56, 65, 66, 67, 68, 69
Offset: 1
Links
- Álvar Ibeas, Table of n, a(n) for n = 1..10000
- Eckford Cohen, A class of residue systems (mod r) and related arithmetical functions. I. A generalization of the Moebius function, Pacific J. Math. 9(1) (1959), 13-24; see Section 6 where it is function Phi_3(m).
- Eckford Cohen, A generalized Euler phi-function, Math. Mag. 41 (1968), 276-279; this is function phi_3(n).
- V. L. Klee, Jr., A generalization of Euler's phi function, Amer. Math. Monthly, 55(6) (1948), 358-359; this is function Phi_3(n).
- Paul J. McCarthy, On a certain family of arithmetic functions, Amer. Math. Monthly 65 (1958), 586-590; this is function T_3(n).
- Franz Rogel, Entwicklung einiger zahlentheoreticher Funktionen in unendliche Reihen, S.-B. Kgl. Bohmischen Ges. Wiss. Article XLVI/XLIV (1897), Prague (26 pages). [This paper deals with arithmetic functions, especially the Euler phi function. It contains interesting generating functions for the function phi. It was continued three years later with the next paper, which contains his function phi_k(n). As stated at the end of the volume, in the table of contents, there is a mistake in numbering the article, so two Roman numerals appear in the literature for labeling this article! - _Petros Hadjicostas_, Jul 21 2019]
- Franz Rogel, Entwicklung einiger zahlentheoreticher Funktionen in unendliche Reihen, S.-B. Kgl. Bohmischen Ges. Wiss. Article XXX (1900), Prague (9 pages). [This is a continuation of the previous article, which was written three years earlier and has the same title. The numbering of the equations continues from the previous paper, but this paper is the one that introduces the function phi_k(n). In our current sequence, k = 3, and a(n) = phi_3(n). Cohen (1959) refers to this paper and correctly attributes this function to F. Rogel. - _Petros Hadjicostas_, Jul 21 2019]
Programs
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Mathematica
f[p_, e_] := If[e < 3, p^e, p^e - p^(e - 3)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)
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PARI
a(n) = {f = factor(n); for (i=1, #f~, if ((e=f[i,2])>=3, f[i,1] = f[i,1]^e - f[i,1]^(e-3); f[i,2]=1);); factorback(f);} \\ Michel Marcus, Feb 10 2015 -
Python
from math import prod from sympy import factorint def A254926(n): return prod(p**e - (p**(e-3) if e >= 3 else 0) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 24 2022
Formula
Multiplicative with a(p^e) = p^e, if e<3; a(p^e) = p^e - p^(e-3), otherwise.
Dirichlet g.f.: zeta(s-1) / zeta(3s).
Sum_{k=1..n} a(k) ~ 945*n^2 / (2*Pi^6). - Vaclav Kotesovec, Feb 02 2019 [This is a special case of a general result by McCarthy (1958), which was re-proved later by Cohen (1968). - Petros Hadjicostas, Jul 20 2019]
a(n) = Sum_{v >= 1} mu(v) * [n, v^3] * (n/v^3), where [n, v^3] = 1 when n is a multiple of v^3, and = 0 otherwise. [This is Eq. (53) in Rogel (1900) and Eq. (6.1) in Cohen (1959).] - Petros Hadjicostas, Jul 21 2019
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} A212793(gcd(n,k)).
a(n) = Sum_{k=1..n} A212793(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
G.f.: Sum_{k>=1} mu(k) * x^(k^3) / (1 - x^(k^3))^2. - Ilya Gutkovskiy, Aug 20 2021
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