

A220171


An ordered subset of primitive values of x^2 + x*y + y^2 where at least two ordered pairs (x1,y1) and (x2,y2) with x1 != x2, y1 != y2 and gcd(x1,y1) = gcd(x2,y2) = 1 yield identical primitive values.


0



91, 133, 217, 247, 259, 273, 301, 399, 403, 427, 469, 481, 511, 553, 559, 589, 637, 651, 679, 703, 721, 741, 763, 777, 793, 817, 871, 889, 903, 931, 949, 973, 1027, 1057, 1099, 1141, 1147, 1159, 1183, 1209, 1261, 1267, 1273, 1281, 1333, 1339, 1351
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OFFSET

1,1


COMMENTS

The primitive values of x^2 + x*y + y^2 where x >= y >= 0 and gcd(x, y) = 1 are given by A034017. However there are incidents in the sequence A034017 where different values of (x, y) yield the same primitive value. Furthermore, the number of solutions for a given primitive value equates to a power of 2. See A121940.


LINKS

Table of n, a(n) for n=1..47.


FORMULA

n such that n = x1^2 + x1*y1 + y1^2 = x2^2 + x2*y2 + y2^2 with x1 != x2, y1 != y2 and gcd(x1,y1) = gcd(x2,y2) = 1.


EXAMPLE

a(3) = 217 because it is the 3rd incident in ascending order of the primitive x^2 + x*y + y^2 that yields multiple solutions. This happens when (x, y) = (9, 8) and (13, 3).


MATHEMATICA

maxLen = 100; sol[k_] := Solve[m^2 + m*n + n^2 == k && m > n > 0 && GCD[m, n] == 1, Integers]; getlist[l_] := Which[Length[sol[l]] == 0, {}, True, {m, n} /. sol[l]]; list = {}; p = 1; While[Length[list] < maxLen, (While[Length[getlist[p]] < 2, p++]; list = Append[list, p]; p++)]; list


CROSSREFS

Cf. A034017, A121940.
Sequence in context: A046427 A179338 A020238 * A260974 A020301 A083517
Adjacent sequences: A220168 A220169 A220170 * A220172 A220173 A220174


KEYWORD

nonn


AUTHOR

Frank M Jackson, Dec 06 2012


STATUS

approved



