This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A050931 #48 Feb 16 2025 08:32:40
%S A050931 7,13,14,19,21,26,28,31,35,37,38,39,42,43,49,52,56,57,61,62,63,65,67,
%T A050931 70,73,74,76,77,78,79,84,86,91,93,95,97,98,103,104,105,109,111,112,
%U A050931 114,117,119,122,124,126,127,129,130,133,134,139,140,143,146,147,148,151
%N A050931 Numbers having a prime factor congruent to 1 mod 6.
%C A050931 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.
%C A050931 Note cot(2*Pi/3) = -1/sqrt(3).
%C A050931 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
%C A050931 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
%C A050931 From _Jean-Christophe Hervé_, Nov 24 2013: (Start)
%C A050931 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.
%C A050931 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.
%C A050931 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).
%C A050931 (End)
%C A050931 Conjecture: Numbers m such that abs(Sum_{k=1..m} [k|m]*A008683(k)*(-1)^(2*k/3)) = 0. - _Mats Granvik_, Jul 06 2024
%H A050931 Reinhard Zumkeller, <a href="/A050931/b050931.txt">Table of n, a(n) for n = 1..10000</a>
%H A050931 Bojan Mohar, <a href="http://arxiv.org/abs/1505.03373">Hermitian adjacency spectrum and switching equivalence of mixed graphs</a>, arXiv preprint arXiv:1505.03373 [math.CO], 2015.
%H A050931 Planet Math, <a href="http://planetmath.org/TruncatedCone">Truncated cone</a>
%H A050931 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Triangle.html">Triangle</a> - see especially (19)
%H A050931 <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%F A050931 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
%F A050931 cot(2*Pi/3) = -1/sqrt(3) = -0.57735... = - A020760. - _M. F. Hasler_, Aug 18 2016
%t A050931 Select[Range[2,200],MemberQ[Union[Mod[#,6]&/@FactorInteger[#][[All,1]]],1]&] (* _Harvey P. Dale_, Aug 24 2019 *)
%o A050931 (Haskell)
%o A050931 a050931 n = a050931_list !! (n-1)
%o A050931 a050931_list = filter (any (== 1) . map (flip mod 6) . a027748_row) [1..]
%o A050931 -- _Reinhard Zumkeller_, Apr 09 2014
%o A050931 (PARI) is_A050931(n)=n>6&&Set(factor(n)[,1]%6)[1]==1 \\ _M. F. Hasler_, Mar 04 2018
%Y A050931 Cf. A002476, A004611, A024606, A230780 (complement), A009003.
%Y A050931 Cf. A027748.
%K A050931 easy,nonn
%O A050931 1,1
%A A050931 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 30 1999
%E A050931 Simpler definition from _M. F. Hasler_, Mar 04 2018