

A254926


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


3



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
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OFFSET

1,2


COMMENTS

Dirichlet convolution of A000010 and A212793.
Möbius transform of A254981.


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), 1324; see Section 6 where it is function Phi_3(m).
Eckford Cohen, A generalized Euler phifunction, Math. Mag. 41 (1968), 276279; this is function phi_3(n).
V. L. Klee, Jr., A generalization of Euler's phi function, Amer. Math. Monthly, 55(6) (1948), 358359; this is function Phi_3(n).
Paul J. McCarthy, On a certain family of arithmetic functions, Amer. Math. Monthly 65 (1958), 586590; 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]


FORMULA

Multiplicative with a(p^e) = p^e, if e<3; a(p^e) = p^e  p^(e3), otherwise.
Dirichlet g.f.: zeta(s1) / 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 reproved 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_{dn} phi(d)*A212793(n/d), where phi = A000010.
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)


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 *)


PROG

(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]^(e3); f[i, 2]=1); ); factorback(f); } \\ Michel Marcus, Feb 10 2015


CROSSREFS

Cf. A000010, A212793, A063659, A254981.
Row 3 of A309287.
Sequence in context: A332713 A017862 A265539 * A155919 A265540 A075011
Adjacent sequences: A254923 A254924 A254925 * A254927 A254928 A254929


KEYWORD

mult,nonn


AUTHOR

Álvar Ibeas, Feb 10 2015


STATUS

approved



