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)
Conjecture: Numbers m such that abs(Sum_{k=1..m} [k|m]*A008683(k)*(-1)^(2*k/3)) = 0. - Mats Granvik, Jul 06 2024
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, Truncated cone
Eric Weisstein's World of Mathematics, Triangle - see especially (19)
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
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