A074232 Positive numbers that are not 3 or 6 (mod 9).

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

Author

Jon Perry, Sep 17 2002

Keywords

• nonn
• easy

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

• Antti Karttunen, Table of n, a(n) for n = 1..12000
• 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.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Crossrefs

Complement of A016051.

Disjoint union of A001651 and A008591.

Cf. A007949, A374039 (characteristic function).

Cf. A327863, A373478, A373992, A374042 (subsequences).

Programs

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 *)

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

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

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