A000164 -id:A000164 - cp's OEIS frontend

A124971 Numbers n which can be expressed as the ordered sum of 3 squares in 2 or more different ways and such that n+1 has the same property.

17, 25, 26, 33, 49, 50, 53, 61, 65, 68, 72, 73, 74, 81, 82, 85, 89, 97, 98, 99, 100, 101, 104, 105, 106, 107, 108, 109, 113, 116, 117, 121, 122, 125, 129, 130, 131, 136, 137, 138, 144, 145, 146, 149, 152, 153, 154, 157, 161, 164, 165, 169, 170, 173, 177, 178

Offset: 1

Artur Jasinski, Nov 14 2006

nonn

			a(1)=17 because 17=3^2+2^2+2^2 = 4^2+1^2+0^2 and a(1)+1= 18=3^2+3^2+0^2 = 4^2+1^2+1^2

R. J. Mathar, Table of n, a(n) for n = 1..6598

Cf. A124966-A124970.

Select[Range[200], Length@PowersRepresentations[#, 3, 2] > 1 && Length@PowersRepresentations[# + 1, 3, 2] > 1 &] (* Ray Chandler, Oct 31 2019 *)

isCnt3sqr(n)={ local(cnt=0,z2) ; for(x=0,floor(sqrt(n)), for(y=x,floor(sqrt(n-x^2)), z2=n-x^2-y^2 ; if( z2>=y^2 && issquare(z2), cnt++ ; ) ; if(cnt >=2, return(1) ) ; ) ; ) ; return(0) ; } isA124971(n)= { return( isCnt3sqr(n) && isCnt3sqr(n+1)) ; } { for(n=1,200, if( isA124971(n), print1(n,", ") ; ) ; ) ; } \\ R. J. Mathar, Nov 29 2006

A000164(n)>=2 and A000164(n+1)>=2. - R. J. Mathar, Nov 29 2006

Corrected and extended by Ray Chandler, Nov 30 2006

Corrected and extended by R. J. Mathar, Nov 29 2006

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

A294578 Numbers which can be expressed as an ordered sum of 3 squares in 6 or more different ways.

Original entry on oeis.org

194, 209, 269, 281, 290, 297, 306, 314, 321, 326, 329, 341, 342, 365, 369, 374, 386, 389, 401, 425, 426, 434, 441, 446, 449, 450, 458, 459, 461, 482, 485, 486, 489, 494, 497, 506, 509, 513, 521, 530, 531, 534, 542, 545, 546, 549, 554, 558, 561, 566, 569, 578

Offset: 1

Robert Price, Nov 02 2017

nonn

Robert Price, Table of n, a(n) for n = 1..80080

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

Select[Range[578], Length[PowersRepresentations[#, 3, 2]] >= 6 &]

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

A321424 Expansion of 1/2 * Product_{0 <= i <= j <= k} (1 + x^(i^2 + j^2 + k^2)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 8, 9, 11, 14, 16, 21, 25, 28, 35, 43, 50, 60, 70, 82, 98, 113, 131, 155, 182, 210, 244, 283, 326, 377, 432, 495, 571, 657, 747, 856, 979, 1112, 1269, 1439, 1632, 1859, 2105, 2377, 2694, 3040, 3426, 3867, 4349, 4894, 5509, 6184, 6939, 7788

Offset: 0

Seiichi Manyama, Nov 09 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000164, A321381, A321423.

G.f.: Product_{k>0} (1 + x^k)^A000164(k).

A224444 Multiplicities for representations of nonnegative numbers as primitive sums of three squares of integers (square 0 allowed).

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Apr 08 2013

nonn

a(n) = 0, for n >= 1, if there is no representation of n as a sum of three squares (square 0 allowed) with no common factor > 1. a(0) = 0 because gcd(0,0,0) = 0 (not 1). a(n) = k >= 1 if n is representable as a primitive sum of three squares (square 0 allowed) in exactly k ways, if neither the order of the three terms nor the signs of the numbers to be squared are taken into account.
Compare with the multiplicities A000164.
The numbers for which a(n) is not 0 are given in A047449.

			a(0) = 0 because  0 = 0^2 + 0^2 + 0^2 is the only candidate for a representation but this is not a primitive sum because gcd(0,0,0) = 0, not 1.
a(2) = 1 because the only candidate for a representation of 2 is the triple [a,b,c] = [0,1,1] and this is primitive, because gcd(0,1,1) = 1.
a(9) = 1 because the two candidate triples are [0, 0, 3] and [1, 2, 2] but [0, 0, 3] is not primitive (gcd(0,0,3) =  3). A000164(9) = 2.
a(17) = 2 with the primitive [a,b,c] triples [0, 1, 4] and [2, 2, 3]. A000164(17) = 2 also.
a(41) = 3 = A000164(41) because the candidate triples [0, 4, 5], [1, 2, 6] and [3, 4, 4] are all primitive.

Cf. A047449, A000164.

Table[ Count[ PowersRepresentations[n, 3, 2], pr_ /; GCD @@ pr == 1], {n, 0, 125}] (* Jean-François Alcover, Apr 09 2013 *)

a(n) = k if n, for n >= 0, has exactly k representations n = a^2 + b^2 + c^2, with a, b and c integers, 0 <= a <= b < = c and gcd(a,b,c) = 1. If there is no such representation a(n) = 0.

A294596 Numbers which can be expressed as an ordered sum of 3 squares in 7 or more different ways.

Original entry on oeis.org

306, 314, 341, 369, 374, 425, 441, 446, 450, 458, 461, 486, 494, 506, 509, 521, 530, 545, 549, 566, 569, 581, 585, 593, 594, 605, 614, 621, 626, 629, 641, 650, 654, 657, 666, 674, 677, 686, 689, 698, 701, 706, 710, 725, 726, 729, 731, 734, 738, 746, 749, 761

Offset: 1

Robert Price, Nov 03 2017

nonn

Robert Price, Table of n, a(n) for n = 1..78680

Cf. A000164, A094942, A124966, A124967, A124968, A124969, A294578.

Select[Range[761], Length[PowersRepresentations[#, 3, 2]] >= 7 &]

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

cp's OEIS Frontend

A124971 Numbers n which can be expressed as the ordered sum of 3 squares in 2 or more different ways and such that n+1 has the same property.

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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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A294578 Numbers which can be expressed as an ordered sum of 3 squares in 6 or more different 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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A321424 Expansion of 1/2 * Product_{0 <= i <= j <= k} (1 + x^(i^2 + j^2 + k^2)).

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