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

1,1

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.

Links

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

Example

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

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 &]

Crossrefs

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

Sequence in context: A250342 A232359 A294714 * A160098 A349940 A203343

Adjacent sequences: A294709 A294710 A294711 * A294713 A294714 A294715

Keyword

nonn

Author

Robert Price, Nov 07 2017

Extensions

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

Status

approved