login
A074232
Positive numbers that are not 3 or 6 (mod 9).
6
1, 2, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 65, 67, 68, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91
OFFSET
1,2
COMMENTS
Previous name was: Numbers n such that Kronecker(9,n) = mu(gcd(9,n)).
From Antti Karttunen, Jun 28 2024: (Start)
Numbers whose 3-adic valuation is not 1; union of non-multiples of 3 and multiples of 9.
A multiplicative semigroup: if m and n are in the sequence, then so is m*n.
(End)
The asymptotic density of this sequence is 7/9. - Amiram Eldar, Jun 28 2024
LINKS
R. D. Carmichael, On the representation of numbers in the form x^3+y^3+z^3-3xyz, Bull. Amer. Math. Soc. 22 (1915), 111-117.
Vladimir Shevelev, Representation of positive integers by the form x^3+y^3+z^3-3xyz, arXiv:1508.05748 [math.NT], 2015.
FORMULA
G.f.: x*(x^2-x+1)*(1+x+x^2)^2 / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Apr 28 2016
MATHEMATICA
Select[Range@ 91, ! Xor[Mod[#, 3] == 0, Mod[#, 9] == 0] &] (* or *)
Select[Range@ 91, KroneckerSymbol[#, 9] == MoebiusMu[GCD[#, 9]] &] (* Michael De Vlieger, Sep 07 2015 *)
PROG
(PARI) lista(nn) = for (n=1, nn, if (kronecker(9, n)==moebius(gcd(9, n)) , print1(n, ", "))); \\ Michel Marcus, Aug 12 2015
(PARI) is(n)=valuation(n, 3)!=1 \\ Charles R Greathouse IV, Aug 12 2015
CROSSREFS
Complement of A016051.
Disjoint union of A001651 and A008591.
Cf. A007949, A374039 (characteristic function).
Cf. A327863, A373478, A373992, A374042 (subsequences).
Sequence in context: A039184 A039137 A071807 * A007417 A039099 A215069
KEYWORD
nonn,easy
AUTHOR
Jon Perry, Sep 17 2002
EXTENSIONS
Offset corrected by Michel Marcus, Aug 12 2015
Definition edited by N. J. A. Sloane, Aug 25 2015
Better name from Vladimir Shevelev, Aug 12 2015
STATUS
approved