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A050931 Numbers having a prime factor congruent to 1 mod 6. 8
7, 13, 14, 19, 21, 26, 28, 31, 35, 37, 38, 39, 42, 43, 49, 52, 56, 57, 61, 62, 63, 65, 67, 70, 73, 74, 76, 77, 78, 79, 84, 86, 91, 93, 95, 97, 98, 103, 104, 105, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 133, 134, 139, 140, 143, 146, 147, 148, 151 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Original definition: Solutions c of cot(2*Pi/3)*(-(a+b+c)*(-a+b+c)*(-a+b-c)*(a+b-c))^(1/2)=a^2+b^2-c^2, c>a,b integers.

Note cot(2*Pi/3) = -1/sqrt(3).

Also the c-values for solutions to c^2 = a^2 + ab + b^2 in positive integers. Also the numbers which occur as the longest side of some triangle with integer sides and a 120-degree angle. - Paul Boddington, Nov 05 2007

The sequence can also be defined as the numbers w which are Heronian means of two distinct positive integers u and v, i.e., w = [u+sqrt(uv)+v]/3. E.g., 28 is the Heronian mean of 4 and 64 (and also of 12 and 48). - Pahikkala Jussi, Feb 16 2008

From Jean-Christophe Hervé, Nov 24 2013: (Start)

This sequence is the analog of hypotenuse numbers A009003 for triangles with integer sides and a 120-degree angle. There are two integers a and b > 0 such that a(n)^2 = a^2 + ab + b^2, and a, b and a(n) are the sides of the triangle: a(n) is the sequence of lengths of the longest side of these triangles. A004611 is the same for primitive triangles.

a and b cannot be equal because sqrt(3) is not rational. Then the values a(n) are such that a(n)^2 is in A024606. It follows that a(n) is the sequence of multiples of primes of form 6k+1 A002476.

The sequence is closed under multiplication. The primitive elements are those with exactly one prime divisor of the form 6k+1 with multiplicity one, which are also those for which there exists a unique 120-degree integer triangle with its longest side equals to a(n).

(End)

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Bojan Mohar, Hermitian adjacency spectrum and switching equivalence of mixed graphs, arXiv preprint arXiv:1505.03373 [math.CO], 2015.

Planet Math, Heronian means

Eric Weisstein's World of Mathematics, Triangle - see especially (19)

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

FORMULA

A005088(a(n)) > 0. Terms are obtained by the products A230780(k)*A004611(p) for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 24 2013

cot(2*Pi/3) = -1/sqrt(3) = -0.57735... = - A020760. - M. F. Hasler, Aug 18 2016

MATHEMATICA

Select[Range[2, 200], MemberQ[Union[Mod[#, 6]&/@FactorInteger[#][[All, 1]]], 1]&] (* Harvey P. Dale, Aug 24 2019 *)

PROG

(Haskell)

a050931 n = a050931_list !! (n-1)

a050931_list = filter (any (== 1) . map (flip mod 6) . a027748_row) [1..]

-- Reinhard Zumkeller, Apr 09 2014

(PARI) is_A050931(n)=n>6&&Set(factor(n)[, 1]%6)[1]==1 \\ M. F. Hasler, Mar 04 2018

CROSSREFS

Cf. A002476, A004611, A024606, A230780 (complement), A009003.

Cf. A027748.

Sequence in context: A057930 A233593 A013651 * A072864 A232437 A275201

Adjacent sequences:  A050928 A050929 A050930 * A050932 A050933 A050934

KEYWORD

easy,nonn

AUTHOR

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 30 1999

EXTENSIONS

Simpler definition from M. F. Hasler, Mar 04 2018

STATUS

approved

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Last modified October 23 17:32 EDT 2019. Contains 328373 sequences. (Running on oeis4.)