A050931 Numbers having a prime factor congruent to 1 mod 6.

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

Offset: 1

Author

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

Keywords

• easy
• nonn

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

Crossrefs

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

Cf. A027748.

Programs

Haskell

a050931 n = a050931_list !! (n-1)
a050931_list = filter (any (== 1) . map (flip mod 6) . a027748_row) [1..]
-- Reinhard Zumkeller, Apr 09 2014

Mathematica

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

PARI

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

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

Extensions

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