A094739 -id:A094739 - cp's OEIS frontend

A337217 One half of the even numbers of A094739.

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

Offset: 1

Wolfdieter Lang, Aug 20 2020

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.

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.

Cf. A002023, A094739, A337218.

A094942 Numbers having a unique partition into three squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 16, 19, 20, 21, 22, 24, 30, 32, 35, 37, 40, 42, 43, 44, 46, 48, 52, 56, 58, 64, 67, 70, 76, 78, 80, 84, 88, 91, 93, 96, 115, 120, 128, 133, 140, 142, 148, 160, 163, 168, 172, 176, 184, 190, 192, 208, 224, 232, 235

Offset: 1

T. D. Noe, May 24 2004

nonn

Note that squares are allowed to be zero.
From Wolfdieter Lang, Apr 09 2013: (Start)
These are the numbers for which A000164(a(n)) = 1.
a(n) is the n-th largest number which has a representation as a sum of three squares (square 0 allowed), in exactly one way, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity with order and signs taken into account are A005875(a(n)).
These numbers are a proper subset of A000378.
(End)
Note that all these numbers are 4^k * A094739(n) for some k >= 0. - T. D. Noe, Nov 08 2013

			From _Wolfdieter Lang_, Apr 09 2013 (Start)
a(1) = 0 because 0 = 0^2 + 0^2 + 0^2 and 0 is the first number m with A000164(m)=1.
a(8) = 8 = 0^2 + 2^2 + 2^2, the 8th largest number m for which A000164(m) is 1.
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A025321 (numbers having a unique partition into three positive squares), A094739 (primitive n having a unique partition into three squares).

Cf. A000164, A005875, A000378, A224442 (two ways), A224443 (three ways).

lim=25; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && nRay Chandler, Oct 31 2019 *)

The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 1, m >= 0}.

0 added by T. D. Noe, Apr 09 2013

A095809 Least positive number having exactly n partitions into three squares.

Original entry on oeis.org

1, 9, 41, 81, 146, 194, 306, 369, 425, 594, 689, 866, 1109, 1161, 1154, 1361, 1634, 1781, 1889, 2141, 2729, 2609, 3626, 3366, 3566, 3449, 3506, 4241, 4289, 4826, 5066, 5381, 7034, 5561, 6254, 7229, 7829, 8186, 8069, 8126, 8609, 8921, 8774, 10386

Offset: 1

T. D. Noe, Jun 07 2004

nonn

Note that a square can be zero.

			41 is the least number having three partitions: 41 = 0+16+25 = 1+4+36 = 9+16+16.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Apart from initial term, same as A000437.

Cf. A094739 (n having a unique partition into three squares), A095811 (greatest number having exactly n partitions into three squares), A124970.

lim=200; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n

A094740 Numbers n such that 4^k n, for k >= 0, are numbers having a unique partition into three positive squares.

Original entry on oeis.org

3, 6, 9, 11, 14, 17, 18, 19, 21, 22, 26, 29, 30, 34, 35, 42, 43, 45, 46, 49, 50, 53, 61, 65, 67, 70, 73, 78, 82, 91, 93, 97, 106, 109, 115, 133, 142, 145, 157, 163, 169, 190, 193, 202, 205, 235, 253, 265, 277, 298, 397, 403, 427, 442, 445, 505, 793

Offset: 1

T. D. Noe, May 24 2004

nonn

It is conjectured that this sequence is complete.

			793 is in this sequence because 793 = 6^2 + 9^2 + 26^2 is the unique partition of 793.

Cf. A094739 (n having a unique partition into three squares).

lim=100; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n0&]

A096017 Numbers n such that 4^k*n, for k >= 0, have a unique partition into three distinct positive squares.

Original entry on oeis.org

14, 21, 26, 29, 30, 35, 38, 41, 42, 45, 46, 49, 50, 53, 54, 59, 61, 65, 66, 70, 75, 78, 81, 83, 91, 93, 106, 107, 109, 113, 114, 115, 118, 121, 133, 137, 139, 142, 145, 147, 153, 157, 162, 169, 171, 178, 190, 198, 202, 205, 211, 214, 219, 226, 235, 243, 253, 258, 262, 265, 277, 283, 289, 291, 298, 307, 313, 323, 331, 337, 358, 363, 379, 387, 397, 403, 418, 427, 438, 442, 445, 457, 466, 498, 499, 505, 547, 562, 577, 603, 643, 723, 793, 883, 907, 1003, 1227, 1243, 1387, 1411, 1467, 1507

Offset: 1

T. D. Noe, Jun 15 2004

It is conjectured that this sequence is complete.

			793 is in this sequence because 793 = 6^2 + 9^2 + 26^2 is the unique partition of 793.

Cf. A094739 (primitive n having a unique partition into three squares), A094740 (primitive n having a unique partition into three positive squares).

lim=100; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n0&]

A337218 The positive integers uniquely represented by the ternary form x^2 + 2y^2 + 2z^2, with integers x <= 0, and 0 <= y <= z.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 12, 13, 14, 21, 22, 30, 37, 42, 46, 48, 58, 70, 78, 93, 133, 142, 190, 192, 253, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888

Offset: 1

Wolfdieter Lang, Aug 20 2020

This sequence gives Theorem 2.2. of Kaplansky, p. 88, with a proof on p. 90.
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.
See eq. (2.2), (2,4), p. 87, of Kaplansky for the two finite sequences with 13 and 8 members, respectively.
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.

			4 is not a member because (x, y, z) = (0, 1, 1) and (2, 0, 0) give both 4.
3 is a member with one solution (1, 0, 1).
5 is a member with one solutuion (1, 1, 1).
7 is not a member because there is no solution.
11 is not a member because there are two solutions (1, 1, 2) and (3, 0, 1).

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.

Cf. A002001, A004771, A094739.

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

A287164 Primes having a unique partition into three squares.

Original entry on oeis.org

2, 3, 5, 11, 13, 19, 37, 43, 67, 163

Offset: 1

Ilya Gutkovskiy, May 20 2017

D. H. Lehmer conjectures that there are no more terms (see A094739 and A094942).

			-------------------------------
|  n | a(n) | representation  |
|-----------------------------|
|  1 |   2  | 0^2 + 1^2 + 1^2 |
|  2 |   3  | 1^2 + 1^2 + 1^2 |
|  3 |   5  | 0^2 + 1^2 + 2^2 |
|  4 |  11  | 1^2 + 1^2 + 3^2 |
|  5 |  13  | 0^2 + 2^2 + 3^2 |
|  6 |  19  | 1^2 + 3^2 + 3^2 |
|  7 |  37  | 0^2 + 1^2 + 6^2 |
|  8 |  43  | 3^2 + 3^2 + 5^2 |
|  9 |  67  | 3^2 + 3^2 + 7^2 |
| 10 | 163  | 1^2 + 9^2 + 9^2 |
-------------------------------
157 is the prime of the form x^2 + y^2 + z^2 with x, y, z >= 0, but is not in the sequence because 157 = 0^2 + 6^2 + 11^2 = 2^2 + 3^2 + 12^2.

Index entries for sequences related to sums of squares

Cf. A000164, A000378, A002313, A042998, A094739, A094942.

Select[Range[200], Length[PowersRepresentations[#, 3, 2]] == 1 && PrimeQ[#] &]

cp's OEIS Frontend

A337217 One half of the even numbers of A094739.

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A094942 Numbers having a unique partition into three squares.

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A095809 Least positive number having exactly n partitions into three squares.

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A094740 Numbers n such that 4^k n, for k >= 0, are numbers having a unique partition into three positive squares.

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A096017 Numbers n such that 4^k*n, for k >= 0, have a unique partition into three distinct positive squares.

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A337218 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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A287164 Primes having a unique partition into three squares.

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A337218 The positive integers uniquely represented by the ternary form x^2 + 2y^2 + 2z^2, with integers x <= 0, and 0 <= y <= z.