A224443 -id:A224443 - cp's OEIS frontend

A094942 Numbers having a unique partition into three squares.

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

A224442 Numbers that are the sum of three squares (square 0 allowed) in exactly two ways.

Original entry on oeis.org

9, 17, 18, 25, 26, 27, 29, 33, 34, 36, 38, 45, 49, 51, 53, 57, 59, 61, 62, 68, 69, 72, 73, 75, 77, 82, 83, 85, 94, 97, 100, 102, 104, 105, 106, 107, 108, 109, 116, 118, 123, 130, 132, 136, 138, 139, 141, 144, 147, 152, 154, 155, 157, 158, 165, 177, 180, 187

Offset: 1

Wolfdieter Lang, Apr 08 2013

nonn

These are the numbers for which A000164(a(n)) = 2.
a(n) is the n-th largest number which has a representation as sum of three integer squares (square 0 allowed), in exactly two ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity for a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

			a(1) = 9 = 0^2 + 0^2 + 3^2 = 1^2 + 2^2 + 2^2, and 9 is the smallest number m with A000164(m) = 2 for m >= 0. The multiplicity with order and signs taken into account is 2*3 + 8*3 = 30 = A005875(9).
The two representations [a,b,c] for a(n), n = 1, ..., 10, are
n=1,   9 = [0, 0, 3], [1, 2, 2],
n=2,  17 = [0, 1, 4], [2, 2, 3],
n=3,  18 = [0, 3, 3], [1, 1, 4],
n=4,  25 = [0, 0, 5], [0, 3, 4],
n=5,  26 = [0, 1, 5], [1, 3, 4],
n=6,  27 = [1, 1, 5], [3, 3, 3],
n=7,  29 = [0, 2, 5], [2, 3, 4],
n=8,  33 = [1, 4, 4], [2, 2, 5],
n=9,  34 = [0, 3, 5], [3, 3, 4],
n=10, 36 = [0, 0, 6], [2, 4, 4].

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000164, A005875, A000378, A094942 (one way), A224443 (three ways).

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^22, 3, min(3, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))))
    end:
a:= proc(n) option remember; local k;
      for k from 1 +`if`(n=1, 0, a(n-1))
      while b(k, isqrt(k), 3)<>2 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 09 2013

Select[ Range[0, 200], Length[ PowersRepresentations[#, 3, 2]] == 2 &] (* Jean-François Alcover, Apr 09 2013 *)

This sequence gives the increasingly ordered elements of the set {m integer | m = a^2 + b^2 + c^2, a, b and c integers with 0 <= a <= b <= c, and m has exactly two such representation}.

The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 2, m >= 0}.

A294577 Numbers that are the sum of three squares (square 0 allowed) in exactly four ways.

Original entry on oeis.org

81, 89, 101, 125, 129, 134, 149, 161, 162, 170, 171, 173, 189, 198, 201, 233, 241, 242, 243, 245, 246, 249, 250, 251, 254, 270, 274, 278, 285, 289, 294, 299, 324, 339, 349, 356, 361, 363, 370, 371, 378, 387, 390, 393, 395, 404, 406, 411, 417, 429, 433, 451

Offset: 1

Robert Price, Nov 02 2017

nonn

These are the numbers for which A000164(a(n)) = 4.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly four 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..1030

Cf. A000164, A005875, A000378, A094942, A224442, A224443.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 4 &]

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

A294594 Numbers that are the sum of three squares (square 0 allowed) in exactly five ways.

Original entry on oeis.org

146, 153, 185, 206, 221, 225, 230, 234, 257, 261, 266, 293, 305, 325, 338, 350, 353, 354, 362, 377, 381, 398, 402, 405, 409, 410, 413, 414, 419, 437, 470, 474, 477, 481, 491, 514, 525, 539, 557, 563, 579, 582, 584, 586, 590, 611, 612, 625, 630, 635, 638, 642

Offset: 1

Robert Price, Nov 03 2017

nonn

These are the numbers for which A000164(a(n)) = 5.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly five 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..843

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 5 &]

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

A294595 Numbers that are the sum of three squares (square 0 allowed) in exactly six ways.

Original entry on oeis.org

194, 209, 269, 281, 290, 297, 321, 326, 329, 342, 365, 386, 389, 401, 426, 434, 449, 459, 482, 485, 489, 497, 513, 531, 534, 542, 546, 554, 558, 561, 578, 601, 602, 633, 649, 659, 662, 665, 675, 678, 681, 693, 699, 705, 713, 714, 722, 737, 741, 747, 750, 754

Offset: 1

Robert Price, Nov 03 2017

nonn

These are the numbers for which A000164(a(n)) = 6.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly six 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..1400

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577, A294594.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 6 &]

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

A294710 Numbers that are the sum of three squares (square 0 allowed) in exactly seven ways.

Original entry on oeis.org

306, 314, 341, 441, 450, 458, 494, 506, 581, 585, 593, 605, 654, 657, 674, 698, 706, 726, 731, 738, 746, 773, 806, 842, 850, 873, 890, 891, 893, 894, 899, 901, 905, 906, 934, 978, 985, 998, 1011, 1013, 1019, 1050, 1058, 1061, 1067, 1073, 1086, 1094, 1101

Offset: 1

Robert Price, Nov 07 2017

nonn

These are the numbers for which A000164(a(n)) = 7.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly seven 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..988

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

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 7 &]

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

A294711 Numbers that are the sum of three squares (square 0 allowed) in exactly eight ways.

Original entry on oeis.org

369, 374, 446, 461, 486, 509, 530, 549, 566, 621, 641, 666, 677, 686, 710, 749, 770, 789, 797, 818, 821, 825, 833, 849, 869, 882, 902, 945, 954, 962, 969, 971, 981, 1010, 1014, 1017, 1022, 1029, 1069, 1085, 1098, 1146, 1157, 1174, 1221, 1233, 1242, 1245

Offset: 1

Robert Price, Nov 07 2017

nonn

These are the numbers for which A000164(a(n)) = 8.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly eight 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..1701

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

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 8 &]

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

A294712 Numbers that are the sum of three squares (square 0 allowed) in exactly nine ways.

Original entry on oeis.org

425, 521, 545, 569, 614, 650, 701, 725, 729, 774, 809, 810, 845, 857, 953, 974, 989, 990, 1053, 1062, 1070, 1074, 1091, 1118, 1134, 1139, 1166, 1179, 1217, 1249, 1251, 1262, 1266, 1277, 1298, 1310, 1418, 1446, 1458, 1470, 1525, 1541, 1546, 1571, 1594, 1611

Offset: 1

Robert Price, Nov 07 2017

nonn

These are the numbers for which A000164(a(n)) = 9.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly nine 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.

			545 =  8^2 + 15^2 + 16^2
    =  0^2 + 16^2 + 17^2
    = 10^2 + 11^2 + 18^2
    =  5^2 + 14^2 + 18^2
    =  8^2 +  9^2 + 20^2
    =  1^2 + 12^2 + 20^2
    =  2^2 + 10^2 + 21^2
    =  5^2 +  6^2 + 22^2
    =  0^2 +  4^2 + 23^2. - _Robert Israel_, Nov 08 2017

Robert Price, Table of n, a(n) for n = 1..1105

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

N:= 10000: # to get all terms <= N
V:= Array(0..N):
for i from 0 to isqrt(N) do
  for j from 0 to i while i^2 + j^2 <= N do
    for k from 0 to j while i^2 + j^2 + k^2 <= N do
      t:= i^2 + j^2 + k^2;
      V[t]:= V[t]+1;
od od od:
select(t -> V[t] = 9, [$1..N]); # Robert Israel, Nov 08 2017

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 9 &]

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

A085625 Numbers that are the sum of 2 squares in exactly 2 ways.

Original entry on oeis.org

25, 50, 65, 85, 100, 125, 130, 145, 169, 170, 185, 200, 205, 221, 225, 250, 260, 265, 289, 290, 305, 338, 340, 365, 370, 377, 400, 410, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 629, 676, 680, 685, 689, 697, 730

Offset: 1

Hugo Pfoertner, Jul 09 2003

nonn

Wells erroneously writes that this sequence begins as 50, 65, 85, 145, ... . - Stefano Spezia, Sep 07 2024

			a(3) = 65 because 65 = 8^2 + 1^2 = 7^2 + 4^2;
a(4) = 85 because 85 = 9^2 + 2^2 = 7^2 + 6^2.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 125.

Robert Price, Table of n, a(n) for n = 1..16794

Cf. A000161, A000446, A224443, A294282, A295153, A295489, A295748.

Select[Range[730], Length[PowersRepresentations[#,2,2]]==2 &] (* Stefano Spezia, Sep 07 2024 *)

n such that A000161(n) = 2.

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

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A094942 Numbers having a unique partition into three squares.

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A224442 Numbers that are the sum of three squares (square 0 allowed) in exactly two ways.

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A294577 Numbers that are the sum of three squares (square 0 allowed) in exactly four ways.

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A294594 Numbers that are the sum of three squares (square 0 allowed) in exactly five ways.

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A294595 Numbers that are the sum of three squares (square 0 allowed) in exactly six ways.

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A294710 Numbers that are the sum of three squares (square 0 allowed) in exactly seven ways.

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A294711 Numbers that are the sum of three squares (square 0 allowed) in exactly eight ways.

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A294712 Numbers that are the sum of three squares (square 0 allowed) in exactly nine ways.

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