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A230780 Positive numbers without a prime factor congruent to 1 (mod 6). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

Jean-Christophe Hervé, Table of n, a(n) for n = 1..254

August Lösch, Economics of Location (1954), see pp. 117f.

U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2

Index entries for sequences related to A2 = hexagonal = triangular lattice

FORMULA

A005088(a(n)) = 0.

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 *)

PROG

(Haskell)

a230780 n = a230780_list !! (n-1)

a230780_list = filter (all (/= 1) . map (flip mod 6) . a027748_row) [1..]

-- Reinhard Zumkeller, Apr 09 2014

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

CROSSREFS

Cf. A002476, A005088, complement of A050931.

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

Cf. A027748.

Sequence in context: A157465 A257247 A269164 * A302540 A067942 A130176

Adjacent sequences:  A230777 A230778 A230779 * A230781 A230782 A230783

KEYWORD

nonn,changed

AUTHOR

Jean-Christophe Hervé, Nov 23 2013

STATUS

approved

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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)