A025427 -id:A025427 - cp's OEIS frontend

A025325 Numbers that are the sum of 3 nonzero squares in exactly 5 ways.

194, 206, 230, 266, 269, 281, 350, 354, 381, 386, 389, 398, 401, 402, 413, 414, 419, 437, 449, 450, 470, 474, 482, 491, 525, 539, 554, 563, 579, 582, 585, 590, 601, 611, 630, 635, 638, 642, 646, 722, 769, 776, 781, 786, 819, 824, 829, 830, 834, 851, 867, 874, 878, 886

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..441
Index entries for sequences related to sums of squares

Cf. A024796, A025427, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338.

A025326 Numbers that are the sum of 3 nonzero squares in exactly 6 ways.

Original entry on oeis.org

209, 297, 306, 314, 321, 326, 329, 342, 425, 426, 434, 441, 458, 459, 489, 497, 513, 530, 531, 534, 542, 546, 558, 561, 593, 602, 605, 633, 649, 650, 657, 659, 662, 665, 674, 675, 678, 681, 693, 698, 699, 705, 706, 713, 714, 725, 737, 738, 741, 746, 747, 750, 755, 758

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..707
Index entries for sequences related to sums of squares

Cf. A024796, A025427, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338.

A025327 Numbers that are the sum of 3 nonzero squares in exactly 7 ways.

Original entry on oeis.org

341, 369, 461, 494, 506, 509, 545, 549, 581, 641, 654, 666, 677, 726, 731, 797, 806, 818, 821, 833, 882, 891, 893, 894, 899, 906, 934, 954, 978, 981, 998, 1011, 1017, 1019, 1050, 1067, 1069, 1086, 1094, 1098, 1101, 1133, 1158, 1194, 1211, 1233, 1294, 1331, 1346

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..422
Index entries for sequences related to sums of squares

Cf. A024796, A025427, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338.

A025328 Numbers that are the sum of 3 nonzero squares in exactly 8 ways.

Original entry on oeis.org

374, 446, 486, 521, 566, 569, 621, 629, 686, 701, 710, 729, 749, 770, 789, 809, 810, 825, 849, 857, 869, 902, 945, 953, 969, 971, 1014, 1022, 1029, 1053, 1085, 1125, 1146, 1174, 1217, 1221, 1241, 1242, 1245, 1249, 1250, 1253, 1254, 1259, 1269, 1277, 1334, 1379

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..649
Index entries for sequences related to sums of squares

Cf. A024796, A025427, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338.

A025329 Numbers that are the sum of 3 nonzero squares in exactly 9 ways.

Original entry on oeis.org

614, 626, 689, 774, 914, 929, 974, 989, 990, 1025, 1062, 1070, 1074, 1091, 1097, 1118, 1134, 1139, 1166, 1179, 1193, 1205, 1229, 1251, 1262, 1266, 1289, 1298, 1305, 1310, 1325, 1409, 1433, 1446, 1470, 1541, 1571, 1611, 1637, 1638, 1745, 1754, 1821, 1834

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..400
Index entries for sequences related to sums of squares

Cf. A024796, A025427, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338.

A025330 Numbers that are the sum of 3 nonzero squares in exactly 10 ways.

Original entry on oeis.org

594, 734, 761, 794, 801, 846, 881, 909, 926, 965, 986, 1001, 1026, 1041, 1089, 1130, 1190, 1209, 1214, 1226, 1265, 1274, 1322, 1326, 1329, 1341, 1370, 1382, 1386, 1505, 1509, 1553, 1557, 1581, 1586, 1613, 1625, 1658, 1689, 1691, 1709, 1713, 1725, 1739

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..592
Index entries for sequences related to sums of squares

Cf. A024796, A025427, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338.

A000419 Numbers that are the sum of 3 but no fewer nonzero squares.

Original entry on oeis.org

3, 6, 11, 12, 14, 19, 21, 22, 24, 27, 30, 33, 35, 38, 42, 43, 44, 46, 48, 51, 54, 56, 57, 59, 62, 66, 67, 69, 70, 75, 76, 77, 78, 83, 84, 86, 88, 91, 93, 94, 96, 99, 102, 105, 107, 108, 110, 114, 115, 118, 120, 123, 126, 129, 131, 132, 133, 134, 138, 139, 140, 141, 142

Offset: 1

N. J. A. Sloane and J. H. Conway

A002828(a(n)) = 3; A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 311.

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A000378, A000408, A000415, A002828, A004215, A025427.

a000419 n = a000419_list !! (n-1)
a000419_list = filter ((== 3) . a002828) [1..]
-- Reinhard Zumkeller, Feb 26 2015

Select[Range[150],SquaresR[3,#]>0&&SquaresR[2,#]==0&] (* Harvey P. Dale, Nov 01 2011 *)

is(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return( n/4^valuation(n,4)%8 !=7 ))); 0 \\ Charles R Greathouse IV, Feb 07 2017

def aupto(lim):
  squares = [k*k for k in range(1, int(lim**.5)+2) if k*k <= lim]
  sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])
  sum3sqs = set(a+b for a in sum2sqs for b in squares)
  return sorted(set(range(lim+1)) & (sum3sqs - sum2sqs - set(squares)))
print(aupto(142)) # Michael S. Branicky, Mar 06 2021

Legendre: a nonnegative integer is a sum of three (or fewer) squares iff it is not of the form 4^k m with m == 7 (mod 8).

More terms from Arlin Anderson (starship1(AT)gmail.com)

A025321 Numbers that are the sum of 3 nonzero squares in exactly 1 way.

Original entry on oeis.org

3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, 26, 29, 30, 34, 35, 36, 42, 43, 44, 45, 46, 48, 49, 50, 53, 56, 61, 65, 67, 68, 70, 72, 73, 76, 78, 82, 84, 88, 91, 93, 96, 97, 104, 106, 109, 115, 116, 120, 133, 136, 140, 142, 144, 145, 157, 163, 168, 169, 172, 176, 180, 184, 190

Offset: 1

David W. Wilson

nonn

It appears that all terms have the form 4^i A094740(j) for some i and j. - T. D. Noe, Jun 06 2008

This is true, because A025427(4*n) = A025427(n) for all n. - Robert Israel, Mar 09 2016

Donovan Johnson, Table of n, a(n) for n = 1..605 (terms < 10^8; first 417 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A000408, A025427, A243148.

lim=20; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && nT. D. Noe, Jun 06 2008 *)
b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<1, 0, b[n, i - 1, k, t] + If[i^2 > n, 0, b[n - i^2, i, k, t - 1]]]];
T[n_, k_] := b[n, Sqrt[n] // Floor, k, k];
Position[Table[T[n, 3], {n, 0, 200}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)

is(n)=if(n<11, return(n>0 && n%3==0)); if(n%4==0, return(is(n/4))); my(w); for(i=sqrtint((n-1)\3)+1,sqrtint(n-2), my(t=n-i^2); for(j=sqrtint((t-1)\2)+1,min(sqrtint(t-1),i), if(issquare(t-j^2), w++>1 && return(0)))); w \\ Charles R Greathouse IV, Aug 05 2024

A243148(a(n),3) = 1. - Alois P. Heinz, Feb 25 2019

A025414 a(n) is the smallest number that is the sum of 3 nonzero squares in exactly n ways.

Original entry on oeis.org

3, 27, 54, 129, 194, 209, 341, 374, 614, 594, 854, 1106, 1314, 1154, 1286, 1746, 1634, 1881, 2141, 2246, 2609, 2889, 3461, 3366, 3449, 3506, 4241, 4289, 5066, 4826, 5381, 5606, 6569, 5561, 6254, 7601, 8186, 8069, 8714, 8126, 9434, 8921, 8774, 11066, 11574

Offset: 1

David W. Wilson

nonn

A025427(a(n)) = n and A025427(m) != n for m < a(n). - Reinhard Zumkeller, Feb 26 2015

			54 is the smallest number having three partitions into nonzero squares: 54 = 1+4+49 = 4+25+25 = 9+9+36.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares

Cf. A094740 (n having a unique partition into three positive squares), A095812 (greatest number having exactly n partitions into three positive squares).

Cf. A025427.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a025414 = fromJust . (`elemIndex` a025427_list)
-- Reinhard Zumkeller, Feb 26 2015

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

A223730 Multiplicities for representations of positive numbers n as primitive sums of three nonzero squares.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 2, 0, 0, 2, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 2, 0, 0, 2, 0, 2, 0, 1, 2, 0, 0, 1, 3, 1, 0, 2, 1, 0, 0, 1, 2, 1, 0, 2, 1, 0, 0, 2, 1, 2, 0, 0, 3, 0, 0, 3, 2, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 3, 2, 0, 0, 2, 1, 2, 0, 1, 3, 0, 0, 2, 3, 1, 0, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 4, 0, 3, 0, 1, 4

Offset: 1

Wolfdieter Lang, Apr 04 2013

nonn

Primitive sums of three nonzero squares a^2 + b^2 + c^2, with positive integers a, b and c, satisfy gcd(a,b,c) = 1. (coprimality of the three squares).
a(n) gives the number of different representations (multiplicities) of the number n >= 1 as primitive sums of three nonzero squares. If a(n) = 0 there is no such representation for n. The numbers n with a(n) not vanishing are given in A223731. The ones with a(n) = 1, 2 and 3 are in A223732, A223733 and A223734, respectively.
For the multiplicities of the positive numbers as sums of three nonzero squares see A025427. The numbers with A025427(n) >= 1 are given in A000408.
A corollary in the Halter-Koch reference (Korollar 1. (b) on p. 13) states for the positive numbers n, not 0, 4, 7 (mod 8) [otherwise n cannot be a primitive sum of three nonzero squares; see p. 11, the r_3(n) formula]:  n is not the sum of three positive coprime squares if and only if n is from the set T := {1, 2, 5, 10, 13, 25, 37, 58, 85, 130, ?}, with ? possibly a number >= 5*10^10 . Therefore a(n) = 0 if and only if n >= 1 is of the form mentioned in this corollary: i)  0, 4, 7 (mod 8) or ii) in the set T.
For representations of n as a sum of three nonzero squares see the Grosswald reference, Theorem 7, p. 79. There also the above mentioned set T appears and for the Conjecture it is assumed that the extra eleventh member of T is absent.

			a(12) = 0 because the only representation of 12 as a sum of three nonzero squares is given by [2,2,2], i.e., 12 = 2^2 + 2^2 + 2^2, but this is not a primitive sum because gcd(2,2,2) = 2, not 1. Such a situation appears for n = 12, 24, 36, 44, 48, 56, 68, 72, 76, 84, 88, 96, ... For these numbers A025427(n) = 1 and a(n) = 0.
a(27) = 1 because the only primitive representation of 27 as a sum of three nonzero squares is denoted by [1,1,5]. The representation [3,3,3] is not primitive.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20.

Cf. A223731, A025427 (non-primitive case), A223732, A223733, A223734.

with(numtheory):
b:= proc(n, i, t, s) option remember;
      `if`(n=0, `if`(t=0 and s={}, 1, 0), `if`(i=1, `if`(t=n, 1, 0),
      `if`(t*i^2xn, 0, b(n-i^2, i, t-1, `if`(s={1}, factorset(i),
       s intersect factorset(i)))))))
    end:
a:= n-> b(n, isqrt(n), 3, {1}):
seq(a(n), n=1..200);  # Alois P. Heinz, Apr 06 2013

a[n_] := Select[ PowersRepresentations[n, 3, 2], Times @@ # != 0 && GCD @@ # == 1 &] // Length; Table[a[n], {n, 1, 134}] (* Jean-François Alcover, Jun 21 2013 *)

a(n) = 0 if there is no representation of n as a primitive sum of three nonzero squares. a(n) = k >= 1 if there are k distinct such representations for n.

cp's OEIS Frontend

A025325 Numbers that are the sum of 3 nonzero squares in exactly 5 ways.

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A025326 Numbers that are the sum of 3 nonzero squares in exactly 6 ways.

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A025327 Numbers that are the sum of 3 nonzero squares in exactly 7 ways.

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A025328 Numbers that are the sum of 3 nonzero squares in exactly 8 ways.

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A025329 Numbers that are the sum of 3 nonzero squares in exactly 9 ways.

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A025330 Numbers that are the sum of 3 nonzero squares in exactly 10 ways.

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A000419 Numbers that are the sum of 3 but no fewer nonzero squares.

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A025321 Numbers that are the sum of 3 nonzero squares in exactly 1 way.

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A025414 a(n) is the smallest number that is the sum of 3 nonzero squares in exactly n ways.

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A223730 Multiplicities for representations of positive numbers n as primitive sums of three nonzero squares.

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