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Numbers having a prime factor congruent to 1 mod 6.
13

%I #47 Jul 06 2024 14:01:57

%S 7,13,14,19,21,26,28,31,35,37,38,39,42,43,49,52,56,57,61,62,63,65,67,

%T 70,73,74,76,77,78,79,84,86,91,93,95,97,98,103,104,105,109,111,112,

%U 114,117,119,122,124,126,127,129,130,133,134,139,140,143,146,147,148,151

%N Numbers having a prime factor congruent to 1 mod 6.

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

%C 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 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 From _Jean-Christophe Hervé_, Nov 24 2013: (Start)

%C 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 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 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 (End)

%C 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 Reinhard Zumkeller, <a href="/A050931/b050931.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 Planet Math, <a href="http://planetmath.org/TruncatedCone">Truncated cone</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Triangle.html">Triangle</a> - see especially (19)

%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>

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

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

%o (Haskell)

%o a050931 n = a050931_list !! (n-1)

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

%o -- _Reinhard Zumkeller_, Apr 09 2014

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

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

%Y Cf. A027748.

%K easy,nonn

%O 1,1

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

%E Simpler definition from _M. F. Hasler_, Mar 04 2018