A000164 -id:A000164 - cp's OEIS frontend

A294597 Numbers which can be expressed as an ordered sum of 3 squares in 8 or more different ways.

369, 374, 425, 446, 461, 486, 509, 521, 530, 545, 549, 566, 569, 594, 614, 621, 626, 629, 641, 650, 666, 677, 686, 689, 701, 710, 725, 729, 734, 749, 761, 770, 774, 789, 794, 797, 801, 809, 810, 818, 821, 825, 833, 845, 846, 849, 854, 857, 866, 869, 881, 882

Offset: 1

Robert Price, Nov 03 2017

nonn

Robert Price, Table of n, a(n) for n = 1..77692

Cf. A000164, A094942, A124966, A124967, A124968.

Select[Range[1000],Length[PowersRepresentations[#,3,2]]>7&] (* Harvey P. Dale, Jul 03 2019 *)

A294713 Numbers that are the sum of three squares (square 0 allowed) in exactly ten ways.

Original entry on oeis.org

594, 626, 629, 734, 846, 914, 926, 929, 1001, 1026, 1041, 1097, 1125, 1190, 1193, 1209, 1214, 1229, 1241, 1265, 1289, 1326, 1329, 1382, 1386, 1409, 1433, 1490, 1505, 1509, 1521, 1530, 1581, 1637, 1689, 1691, 1713, 1725, 1730, 1739, 1749, 1754, 1770, 1778

Offset: 1

Robert Price, Nov 07 2017

nonn

These are the numbers for which A000164(a(n)) = 10.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly ten ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

Robert Price, Table of n, a(n) for n = 1..1835

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577, A294594, A294595, A294710, A294711, A294712.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 10 &]

Updated Mathematica program to Version 11. by Robert Price, Nov 01 2019

A294714 Numbers which can be expressed as an ordered sum of 3 squares in 9 or more different ways.

Original entry on oeis.org

425, 521, 545, 569, 594, 614, 626, 629, 650, 689, 701, 725, 729, 734, 761, 774, 794, 801, 809, 810, 845, 846, 854, 857, 866, 881, 909, 914, 926, 929, 941, 950, 953, 965, 974, 986, 989, 990, 1001, 1025, 1026, 1034, 1041, 1046, 1049, 1053, 1062, 1070, 1074

Offset: 1

Robert Price, Nov 07 2017

nonn

Robert Price, Table of n, a(n) for n = 1..75991

Cf. A000164, A094942, A124966, A124967, A124968.

Select[Range[882], Length[PowersRepresentations[#, 3, 2]] >= 9 &]

Updated Mathematica program to Version 11. by Robert Price, Nov 01 2019

A297930 Number of partitions of n into 2 squares and 2 nonnegative cubes.

Original entry on oeis.org

1, 2, 3, 2, 2, 2, 2, 1, 2, 4, 5, 3, 2, 3, 2, 1, 3, 5, 6, 3, 3, 3, 2, 0, 2, 5, 6, 5, 4, 5, 2, 2, 4, 5, 6, 4, 6, 6, 4, 2, 4, 6, 4, 4, 4, 7, 3, 2, 4, 3, 5, 4, 7, 8, 5, 3, 3, 3, 5, 5, 5, 6, 4, 3, 6, 7, 8, 7, 5, 7, 4, 2, 7, 9, 10, 4, 5, 7, 3, 3, 9, 10, 8, 5, 4, 7

Offset: 0

XU Pingya, Jan 08 2018

nonn

For n <= 6 * 10^7, except for a(23) = 0, all a(n) > 0.

First occurrence of k beginning with 0: 23, 7, 1, 2, 9, 10, 18, 45, 53, 73, 74, 101, 125, 146, 165, 197, ..., . - Robert G. Wilson v, Jan 14 2018

			2 = 0^2 + 0^2 + 1^3 + 1^3 = 0^2 + 1^2 + 0^3 + 1^3 = 1^2 + 1^2 + 0^3 + 0^3, a(2) = 3.
10 = 0^2 + 1^2 + 1^3 + 2^3 = 0^2 + 3^2 + 0^3 + 1^3 = 1^2 + 1^2 + 0^3 + 2^3 = 1^2 + 3^2 + 0^3 + 0^3 = 2^2 + 2^2 + 1^3 + 1^3, a(10) = 5.

W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995) pp. 169-173.
Eric Weisstein's World of Mathematics, Waring's Problem

Cf. A000164, A002635, A022552, A274274.

a[n_] := Sum[If[x^2 + y^2 + z^3 + u^3 == n, 1, 0], {x, 0, n^(1/2)}, {y, x, (n - x^2)^(1/2)}, {z, 0, (n - x^2 - y^2)^(1/3)}, {u, z, (n - x^2 - y^2 - z^3)^(1/3)}]; Table[a[n], {n, 0, 86}]

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[#] &]

A294081 Number of partitions of n into three squares and two nonnegative 7th powers.

Original entry on oeis.org

1, 2, 3, 3, 3, 3, 3, 2, 2, 3, 4, 4, 3, 3, 3, 2, 2, 3, 5, 5, 4, 3, 3, 2, 2, 3, 5, 6, 4, 4, 3, 3, 2, 3, 5, 5, 5, 4, 5, 3, 3, 4, 5, 5, 3, 4, 4, 3, 2, 3, 6, 7, 6, 5, 6, 5, 4, 3, 4, 5, 3, 4, 4, 4, 3, 4, 7, 7, 6, 5, 5, 3, 3, 4, 7, 7, 6, 5, 4, 3, 2, 5, 7, 8, 5, 5, 6

Offset: 0

XU Pingya, Feb 09 2018

nonn

4^i(8j + 7) - 1^7 - 1^7 == 5 (mod 8) (when i = 0), or 2 (when i = 1), or 6 (when i >= 2). Thus, each nonnegative integer can be written as a sum of three squares and two nonnegative 7th powers; i.e., a(n) > 0.

More generally, each nonnegative integer can be written as a sum of three squares and a nonnegative k-th power and a nonnegative m-th power.

			7 = 0^2 + 1^2 + 2^2 + 1^7 + 1^7 = 1^2 + 1^1 + 2^2 + 0^7 + 1^7, a(7) = 2.
10 = 0^2 + 0^2 + 3^2 + 0^7 + 1^7 = 0^2 + 1^1 + 3^2 + 0^7 + 0^7 = 0^2 + 2^2 + 2^2 + 1^7 + 1^7 = 1^2 + 2^1 + 2^2 + 0^7 + 1^7, a(10) = 4.

Wikipedia, Legendre's three-square theorem
Wikipedia, Lagrange's four-square theorem

Cf. A000164, A002635, A004215, A004771, A297970.

a[n_]:=Sum[If[x^2+y^2+z^2+u^7+v^7==n, 1, 0], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {u,0,(n-x^2-y^2-z^2)^(1/7)}, {v,u,(n-x^2-y^2-z^2-u^7)^(1/7)}]
Table[a[n], {n,0,86}]

A294715 Numbers which can be expressed as an ordered sum of 3 squares in 10 or more different ways.

Original entry on oeis.org

594, 626, 629, 689, 734, 761, 794, 801, 846, 854, 866, 881, 909, 914, 926, 929, 941, 950, 965, 986, 1001, 1025, 1026, 1034, 1041, 1046, 1049, 1089, 1097, 1106, 1109, 1121, 1125, 1130, 1154, 1161, 1169, 1181, 1190, 1193, 1205, 1206, 1209, 1214, 1226, 1229

Offset: 1

Robert Price, Nov 07 2017

nonn

Cf. A000164, A094942, A124966, A124967, A124968.

Select[Range[1229], Length[PowersRepresentations[#, 3, 2]] >= 10 &]

Updated Mathematica program to Version 11, and corrected errors in Name. by Robert Price, Nov 01 2019

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A294597 Numbers which can be expressed as an ordered sum of 3 squares in 8 or more different ways.

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A294713 Numbers that are the sum of three squares (square 0 allowed) in exactly ten ways.

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A294714 Numbers which can be expressed as an ordered sum of 3 squares in 9 or more different ways.

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A297930 Number of partitions of n into 2 squares and 2 nonnegative cubes.

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A287164 Primes having a unique partition into three squares.

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A294081 Number of partitions of n into three squares and two nonnegative 7th powers.

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Mathematica

A294715 Numbers which can be expressed as an ordered sum of 3 squares in 10 or more different ways.

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Author

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Mathematica

Extensions