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A254926 There are a(n) numbers m such that 1 <= m <= n and gcd(m,n) is cubefree.

%I A254926 #60 Jan 24 2022 15:58:18
%S A254926 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,
%T A254926 26,28,29,30,31,28,33,34,35,36,37,38,39,35,41,42,43,44,45,46,47,42,49,
%U A254926 50,51,52,53,52,55,49,57,58,59,60,61,62,63,56,65,66,67,68,69
%N A254926 There are a(n) numbers m such that 1 <= m <= n and gcd(m,n) is cubefree.
%C A254926 Dirichlet convolution of A000010 and A212793.
%C A254926 Möbius transform of A254981.
%H A254926 Álvar Ibeas, <a href="/A254926/b254926.txt">Table of n, a(n) for n = 1..10000</a>
%H A254926 Eckford Cohen, <a href="https://msp.org/pjm/1959/9-1/pjm-v9-n1-s.pdf">A class of residue systems (mod r) and related arithmetical functions. I. A generalization of the Moebius function</a>, Pacific J. Math. 9(1) (1959), 13-24; see Section 6 where it is function Phi_3(m).
%H A254926 Eckford Cohen, <a href="https://www.jstor.org/stable/2688817">A generalized Euler phi-function</a>, Math. Mag. 41 (1968), 276-279; this is function phi_3(n).
%H A254926 V. L. Klee, Jr., <a href="https://www.jstor.org/stable/2304963">A generalization of Euler's phi function</a>, Amer. Math. Monthly, 55(6) (1948), 358-359; this is function Phi_3(n).
%H A254926 Paul J. McCarthy, <a href="https://www.jstor.org/stable/2309112">On a certain family of arithmetic functions</a>, Amer. Math. Monthly 65 (1958), 586-590; this is function T_3(n).
%H A254926 Franz Rogel, <a href="https://www.zobodat.at/pdf/SB-Ges-Wiss-Prag_1896_2_0001-0687.pdf">Entwicklung einiger zahlentheoreticher Funktionen in unendliche Reihen</a>, 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]
%H A254926 Franz Rogel, <a href="https://archive.org/details/vestnkkrlovs1900kn/page/n863">Entwicklung einiger zahlentheoreticher Funktionen in unendliche Reihen</a>, 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]
%F A254926 Multiplicative with a(p^e) = p^e, if e<3; a(p^e) = p^e - p^(e-3), otherwise.
%F A254926 Dirichlet g.f.: zeta(s-1) / zeta(3s).
%F A254926 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]
%F A254926 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
%F A254926 From _Richard L. Ollerton_, May 07 2021: (Start)
%F A254926 a(n) = Sum_{d|n} phi(d)*A212793(n/d), where phi = A000010.
%F A254926 a(n) = Sum_{k=1..n} A212793(gcd(n,k)).
%F A254926 a(n) = Sum_{k=1..n} A212793(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
%F A254926 G.f.: Sum_{k>=1} mu(k) * x^(k^3) / (1 - x^(k^3))^2. - _Ilya Gutkovskiy_, Aug 20 2021
%t A254926 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 *)
%o A254926 (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
%o A254926 (Python)
%o A254926 from math import prod
%o A254926 from sympy import factorint
%o A254926 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
%Y A254926 Cf. A000010, A212793, A063659, A254981.
%Y A254926 Row 3 of A309287.
%K A254926 mult,nonn
%O A254926 1,2
%A A254926 _Álvar Ibeas_, Feb 10 2015