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The positive integers uniquely represented by the ternary form x^2 + 2*y^2 + 2*z^2, with integers x <= 0, and 0 <= y <= z.
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%I #6 Aug 24 2020 22:54:11

%S 1,2,3,5,6,10,12,13,14,21,22,30,37,42,46,48,58,70,78,93,133,142,190,

%T 192,253,768,3072,12288,49152,196608,786432,3145728,12582912,50331648,

%U 201326592,805306368,3221225472,12884901888

%N The positive integers uniquely represented by the ternary form x^2 + 2*y^2 + 2*z^2, with integers x <= 0, and 0 <= y <= z.

%C This sequence gives Theorem 2.2. of Kaplansky, p. 88, with a proof on p. 90.

%C This sequence is composed of two finite ones and an infinite one: (i) 2*A337217 = {2, 6, 10, 14, 22, 30, 42, 46, 58, 70, 78, 142, 190}, the even members of A094739, (ii) {1, 5, 13, 21, 37, 93, 133, 253}, the 1 (mod 4) members of A094739, and (iii) A002001(k+1) = 4^k*3, for integer k >= 0. Beginning with a(26) = 768 only the powers 4^k*3, for k >= 4 appear.

%C See eq. (2.2), (2,4), p. 87, of Kaplansky for the two finite sequences with 13 and 8 members, respectively.

%C The positive integers which have no such solution (x, y, z) are given by 4^k*(7+8*m) = A002001(k+1)*A004771(m), for k >= 0 and m >= 0. See Kaplansky, p. 88. The other missing positive integers have more than 1 solution.

%D Irving Kaplansky, Integers Uniquely Represented by Certain Ternary Forms, in "The Mathematics of Paul Erdős I", Ronald. L. Graham and Jaroslav Nešetřil (Eds.), Springer, 1997, pp. 86 - 94.

%F See the comment for the union of the three sequences (i), (ii) and (iii).

%e 4 is not a member because (x, y, z) = (0, 1, 1) and (2, 0, 0) give both 4.

%e 3 is a member with one solution (1, 0, 1).

%e 5 is a member with one solutuion (1, 1, 1).

%e 7 is not a member because there is no solution.

%e 11 is not a member because there are two solutions (1, 1, 2) and (3, 0, 1).

%Y Cf. A002001, A004771, A094739.

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Aug 20 2020