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

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

Author

Robert Price, Nov 07 2017

Keywords

• nonn

Comments

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.

Examples

			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
		

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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

Extensions

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