A337217 One half of the even numbers of A094739.

1, 3, 5, 7, 11, 15, 21, 23, 29, 35, 39, 71, 95

Offset: 1

Author

Wolfdieter Lang, Aug 20 2020

Keywords

• nonn
• fini
• full

Comments

This finite sequence a(n), for n = 1, 2, ..., 13, appears as eq. (2.3) given by Kaplansky on p. 87.

It enters Theorem 2.1 of Kaplansky, p. 87, with proof on p. 90 (here reformulated): The positive integers uniquely represented by x^2 + y^2 + 2*z^2, with 0 <= x <= y and 0 <= z, consist of the 13 numbers a(n) and 4^k*6 = A002023(k), for integers k >= 0. See a comment in A002023 for this uniquely representable positive integers of this ternary form.

It also enters Theorem 2.3 of Kaplansky, p. 88, with proof on p.91 (here reformulated): The positive integers uniquely represented by x^2 + 2*y^2 + 4*z^2, with nonnegative integers x, y, z consist of the 13 odd numbers a(n) and the four even numbers 2, 10, 26, and 74. This is the finite sequence

1, 2, 3, 5, 7, 10, 11, 15, 21, 23, 26, 29, 35, 39, 71, 74, 95.

References

• 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.

Crossrefs

Cf. A002023, A094739, A337218.