A024796 -id:A024796 - cp's OEIS frontend

A306396 Consider the numbers in A024796, numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 <= i <= j <= k; sequence number of ways these numbers can be expressed.

2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 4, 3, 2, 4, 2, 2, 2, 2, 4, 2, 3, 3, 2, 4, 2, 2, 2, 4, 3, 2, 2, 3, 2, 4, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 4, 5, 2, 2, 4, 4, 2, 2, 5, 6, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 2, 2, 5, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 2, 4, 2, 4, 4, 4, 3, 2, 4, 2, 3, 5, 2, 5, 4, 2

Offset: 1

A. Timothy Royappa, Feb 12 2019

nonn

Number of accidental degeneracies in the quantum mechanical 3-D "particle-in-a-box" model.

			The fourth term in A024796 is 41, which can be expressed in two ways as the sum of three nonzero squares (1^2 + 2^2 + 6^2 or 3^2 + 4^2 + 4^2), so a(4) = 2.

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A024796, A025427.

r[n_] := Length@ IntegerPartitions[n, {3}, Range[Sqrt[n]]^2]; Select[ Array[r, 300], # > 1 &] (* Giovanni Resta, Feb 21 2020 *)

a(n) = A025427(A024796(n)).

Offset changed to 1 by Jinyuan Wang, Feb 20 2020

A000408 Numbers that are the sum of three nonzero squares.

Original entry on oeis.org

3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, 26, 27, 29, 30, 33, 34, 35, 36, 38, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 56, 57, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 86, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 101, 102, 104

Offset: 1

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

nonn

a(n) !== 7 (mod 8). - Boris Putievskiy, May 05 2013
A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015
According to Halter-Koch (below), a number n is a sum of 3 squares, but not a sum of 3 nonzero squares (i.e., is in A000378 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1,2,5,10,13,25,37,58,85,130,?}, where ? denotes at most one unknown number that, if it exists, is > 5*10^10. - Jeffrey Shallit, Jan 15 2017

L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Dover, 2005, p. 267.
Savin Réalis, Answer to question 25 ("Toute puissance entière de 3 est une somme de trois carrés premiers avec 3"), Mathesis 1 (1881), pp. 87-88. (See also p. 73 where the question is posed.)

Ray Chandler, Table of n, a(n) for n = 1..10000
Alexander Berkovich and Will Jagy, On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2, arXiv:1101.2951 [math.NT], 2011.
B. Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), J. Int. Seq. 16 (2013) #13.6.4.
Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), pp. 11-20.
S. Mezroui, A. Azizi, and M'hammed Ziane, On a Conjecture of Farhi , J. Int. Seq. 17 (2014) #14.1.8.
Index entries for sequences related to sums of squares

Cf. A000378, A000404, A000414, A003072, A004214, A024795, A024796, A025321, A025427.

a000408 n = a000408_list !! (n-1)
a000408_list = filter ((> 0) . a025427) [1..]
-- Reinhard Zumkeller, Feb 26 2015

N:= 1000: # to get all terms <= N
S:= series((JacobiTheta3(0,q)-1)^3,q,1001):
select(t -> coeff(S,q,t)>0, [$1..N]); # Robert Israel, Jan 14 2016

f[n_] := Flatten[Position[Take[Rest[CoefficientList[Sum[x^(i^2), {i, n}]^3, x]], n^2], ?Positive]];f[11] (* _Ray Chandler, Dec 06 2006 *)
pr[n_] := Select[ PowersRepresentations[n, 3, 2], FreeQ[#, 0] &]; Select[ Range[104], pr[#] != {} &] (* Jean-François Alcover, Apr 04 2013 *)
max = 1000; s = (EllipticTheta[3, 0, q] - 1)^3 + O[q]^(max+1); Select[ Range[max], SeriesCoefficient[s, {q, 0, #}] > 0 &] (* Jean-François Alcover, Feb 01 2016, after Robert Israel *)

is(n)=for(x=sqrtint((n-1)\3)+1,sqrtint(n-2), for(y=1,sqrtint(n-x^2-1), if(issquare(n-x^2-y^2), return(1)))); 0 \\ Charles R Greathouse IV, Apr 04 2013

is(n)= my(a, b) ; a=1 ; while(a^2+1Altug Alkan, Jan 18 2016

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)
print(aupto(104)) # Michael S. Branicky, Mar 06 2021

a(n) = 6n/5 + O(log n). - Charles R Greathouse IV, Mar 14 2014; error term improved Jul 05 2024

A025333 Numbers that are the sum of 3 nonzero squares in 5 or more ways.

Original entry on oeis.org

194, 206, 209, 230, 266, 269, 281, 297, 306, 314, 321, 326, 329, 341, 342, 350, 354, 369, 374, 381, 386, 389, 398, 401, 402, 413, 414, 419, 425, 426, 434, 437, 441, 446, 449, 450, 458, 459, 461, 470, 474, 482, 486, 489, 491, 494, 497, 506, 509, 513, 521, 525, 530, 531

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..6628
Index entries for sequences related to sums of squares

Cf. A024796, A025296, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025370, A025427, A343967.

A025332 Numbers that are the sum of 3 nonzero squares in 4 or more ways.

Original entry on oeis.org

129, 134, 146, 153, 161, 171, 189, 194, 198, 201, 206, 209, 230, 234, 243, 246, 249, 251, 254, 257, 261, 266, 269, 270, 278, 281, 285, 290, 293, 294, 297, 299, 306, 314, 321, 326, 329, 339, 341, 342, 350, 353, 354, 362, 363, 365, 369, 371, 374, 378, 381, 386, 387, 389

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..7221
Index entries for sequences related to sums of squares

Cf. A024796, A025295, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025369, A025427, A343968.

A025334 Numbers that are the sum of 3 nonzero squares in 6 or more ways.

Original entry on oeis.org

209, 297, 306, 314, 321, 326, 329, 341, 342, 369, 374, 425, 426, 434, 441, 446, 458, 459, 461, 486, 489, 494, 497, 506, 509, 513, 521, 530, 531, 534, 542, 545, 546, 549, 558, 561, 566, 569, 581, 593, 594, 602, 605, 614, 621, 626, 629, 633, 641, 649, 650, 654, 657, 659

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..6187
Index entries for sequences related to sums of squares

Cf. A024796, A025297, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025371, A025427, A345083.

A025335 Numbers that are the sum of 3 nonzero squares in 7 or more ways.

Original entry on oeis.org

341, 369, 374, 446, 461, 486, 494, 506, 509, 521, 545, 549, 566, 569, 581, 594, 614, 621, 626, 629, 641, 654, 666, 677, 686, 689, 701, 710, 726, 729, 731, 734, 749, 761, 770, 774, 789, 794, 797, 801, 806, 809, 810, 818, 821, 825, 833, 846, 849, 854, 857, 866, 869, 881

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..5480
Index entries for sequences related to sums of squares

Cf. A024796, A025298, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025372, A025427, A345086.

A025336 Numbers that are the sum of 3 nonzero squares in 8 or more ways.

Original entry on oeis.org

374, 446, 486, 521, 566, 569, 594, 614, 621, 626, 629, 686, 689, 701, 710, 729, 734, 749, 761, 770, 774, 789, 794, 801, 809, 810, 825, 846, 849, 854, 857, 866, 869, 881, 902, 909, 914, 926, 929, 941, 945, 950, 953, 965, 969, 971, 974, 986, 989, 990, 1001, 1014, 1022

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..5058
Index entries for sequences related to sums of squares

Cf. A024796, A025299, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025373, A025427, A345087.

A025337 Numbers that are the sum of 3 nonzero squares in 9 or more ways.

Original entry on oeis.org

594, 614, 626, 689, 734, 761, 774, 794, 801, 846, 854, 866, 881, 909, 914, 926, 929, 941, 950, 965, 974, 986, 989, 990, 1001, 1025, 1026, 1034, 1041, 1046, 1049, 1062, 1070, 1074, 1089, 1091, 1097, 1106, 1109, 1118, 1121, 1130, 1134, 1139, 1154, 1161, 1166

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..4409
Index entries for sequences related to sums of squares

Cf. A024796, A025300, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025374, A025427, A345119.

A025338 Numbers that are the sum of 3 nonzero squares in 10 or more ways.

Original entry on oeis.org

594, 734, 761, 794, 801, 846, 854, 866, 881, 909, 926, 941, 950, 965, 986, 1001, 1026, 1034, 1041, 1046, 1049, 1089, 1106, 1109, 1121, 1130, 1154, 1161, 1169, 1181, 1190, 1206, 1209, 1214, 1226, 1238, 1265, 1274, 1286, 1301, 1314, 1322, 1326, 1329, 1341

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..4009
Index entries for sequences related to sums of squares

Cf. A024796, A025301, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025375, A025427, A345121.

A025331 Numbers that are the sum of 3 nonzero squares in 3 or more ways.

Original entry on oeis.org

54, 66, 81, 86, 89, 99, 101, 110, 114, 126, 129, 131, 134, 146, 149, 150, 153, 161, 162, 166, 171, 173, 174, 179, 182, 185, 186, 189, 194, 198, 201, 206, 209, 216, 219, 221, 222, 225, 227, 230, 233, 234, 237, 241, 242, 243, 245, 246, 249, 251, 254, 257, 258, 261, 264

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..7639
Index entries for sequences related to sums of squares

Cf. A024796, A025294, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025368, A025398, A025427.

cp's OEIS Frontend

A306396 Consider the numbers in A024796, numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 <= i <= j <= k; sequence number of ways these numbers can be expressed.

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A000408 Numbers that are the sum of three nonzero squares.

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A025333 Numbers that are the sum of 3 nonzero squares in 5 or more ways.

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A025332 Numbers that are the sum of 3 nonzero squares in 4 or more ways.

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A025334 Numbers that are the sum of 3 nonzero squares in 6 or more ways.

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A025335 Numbers that are the sum of 3 nonzero squares in 7 or more ways.

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A025336 Numbers that are the sum of 3 nonzero squares in 8 or more ways.

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A025337 Numbers that are the sum of 3 nonzero squares in 9 or more ways.

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A025338 Numbers that are the sum of 3 nonzero squares in 10 or more ways.

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A025331 Numbers that are the sum of 3 nonzero squares in 3 or more ways.

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