A230780 Positive numbers without a prime factor congruent to 1 (mod 6).

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 17, 18, 20, 22, 23, 24, 25, 27, 29, 30, 32, 33, 34, 36, 40, 41, 44, 45, 46, 47, 48, 50, 51, 53, 54, 55, 58, 59, 60, 64, 66, 68, 69, 71, 72, 75, 80, 81, 82, 83, 85, 87, 88, 89, 90, 92, 94, 96, 99, 100, 101, 102, 106, 107

Offset: 1

Author

Jean-Christophe Hervé, Nov 23 2013

Keywords

• nonn

Comments

The sequence is closed under multiplication. Primitive elements are 3 and the primes of form 3*k+2.

a(n)^2 is not expressible as x^2+xy+y^2 with x and y positive integers.

Analog of A004144 (nonhypotenuse numbers) for 120-degree angle triangles: a(n) is not the length of the longest side of such a triangle with integer sides.

It might have been natural to include 0 in this sequence. - M. F. Hasler, Mar 04 2018

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 254 terms from Jean-Christophe Hervé)
• F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
• August Lösch, Economics of Location (1954), see pp. 117f.
• U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
• Index entries for sequences related to A2 = hexagonal = triangular lattice

Crossrefs

Cf. A002476, A005088, complement of A050931.

Cf. A004144 (analog for 4k+1 primes and right triangles).

Cf. A027748.

Programs

Haskell

a230780 n = a230780_list !! (n-1)
a230780_list = filter (all (/= 1) . map (flip mod 6) . a027748_row) [1..]
-- Reinhard Zumkeller, Apr 09 2014

Mathematica

Join[{1}, Select[Range[2, 110], ! MemberQ[Union[Mod[Transpose[ FactorInteger[#]][[1]], 6]], 1] &]] (* T. D. Noe, Nov 24 2013 *)
Join[{1},Select[Range[110],NoneTrue[FactorInteger[#][[All,1]],Mod[#,6] == 1&]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 03 2019 *)

PARI

is_A230780(n)=!setsearch(Set(factor(n)[,1]%6),1) \\ M. F. Hasler, Mar 04 2018

Formula

A005088(a(n)) = 0.