A230780 -id:A230780 - cp's OEIS frontend

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

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

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

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

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

Cf. A027748.

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

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

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

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

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

A232437 Numbers whose square is expressible in only one way as x^2+xy+y^2, with x and y > 0.

Original entry on oeis.org

7, 13, 14, 19, 21, 26, 28, 31, 35, 37, 38, 39, 42, 43, 52, 56, 57, 61, 62, 63, 65, 67, 70, 73, 74, 76, 77, 78, 79, 84, 86, 93, 95, 97, 103, 104, 105, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 134, 139, 140, 143, 146, 148, 151, 152, 154, 155, 156, 157, 158, 161

Offset: 1

Jean-Christophe Hervé, Nov 24 2013

nonn

Analog of A084645 for 120-degree angle triangles with integer sides.
Numbers with exactly one prime divisor of the form 6k+1 with multiplicity one.
Primitive elements of A050931.

			a(1) = 7 as 7^2 = 3^2 + 3*5 + 5^2.

Cf. A002476, A050931, A230780, A232436 (subsequence).

Cf. A084645, A232437, A248599, A254063, A254064.

r[k_] := Reduce[x>0 && y>0 && k^2 == x^2 + x y + y^2, {x, y}, Integers];
selQ[k_] := Which[rk = r[k]; rk === False, False, rk[[0]] === And && Length[rk] == 2, False, rk[[0]] === Or && Length[rk] == 2, True, True, False];
Select[Range[1000], selQ] (* Jean-François Alcover, Feb 20 2020 *)

Terms are obtained by the products A230780(k)*A002476(p) for k, p > 0, ordered by increasing values.

A300303 Squares that are not of the form x^2 + x*y + y^2, where x and y are positive integers.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 64, 81, 100, 121, 144, 225, 256, 289, 324, 400, 484, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2809, 2916, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761, 5041, 5184, 5625, 6400, 6561, 6724, 6889, 7225, 7569

Offset: 1

Altug Alkan, Mar 02 2018

Or Loeschian numbers (A003136) that are not in A024614.
Squares that are not in this sequence are 49, 169, 196, 361, 441, 676, ...
This is the list of squares not of the form A050931(k)^2. A number n is in this sequence iff n = m^2 with m having no prime factor == 1 (mod 6). - M. F. Hasler, Mar 04 2018

			Loeschian number 25 = 5^2 is a term because 25 = x^2 + x*y + y^2 has no solution for positive integers x, y.

Cf. A000548, A003136, A024614, A050931, A230780.

isA024614:= proc(n) local x,y;
for x from 1 to floor(sqrt(n-1)) do
   if issqr(4*n-3*x^2) then return true fi
od:
false
end proc:
isA024614(0):= false:
remove(isA024614, [seq(i^2,i=0..200)]); # Robert Israel, Mar 02 2018

sol[s_] := Solve[0 < x <= y && s == x^2 + x y + y^2, {x, y}, Integers];
Select[Range[0, 100]^2, sol[#] == {}&] (* Jean-François Alcover, Oct 26 2020 *)

is(n,m)=issquare(n,m)&&!setsearch(Set(factor(m)[,1]%6),1) \\ second part is equivalent to is_A230780(m), this is sufficient to test (e.g., to produce a list) if we know that n = m^2. - M. F. Hasler, Mar 04 2018

a(n) = A230780(n-1)^2 for n > 1.

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

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A232437 Numbers whose square is expressible in only one way as x^2+xy+y^2, with x and y > 0.

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A300303 Squares that are not of the form x^2 + x*y + y^2, where x and y are positive integers.

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Keywords

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Examples

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Programs

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Mathematica

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