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A224443
Numbers that are the sum of three squares (square 0 allowed) in exactly three ways.
11
41, 50, 54, 65, 66, 74, 86, 90, 98, 99, 110, 113, 114, 117, 121, 122, 126, 131, 137, 145, 150, 164, 166, 169, 174, 178, 179, 181, 182, 186, 197, 200, 205, 216, 218, 219, 222, 226, 227, 229, 237, 258, 260, 264, 265, 275, 286, 291, 296, 302
OFFSET
1,1
COMMENTS
These are the numbers for which A000164(a(n)) = 3.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly three 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
FORMULA
This sequence gives the increasingly ordered numbers of the set {m integer | m = a^2 + b^2 + c^2, a, b and c integers with 0 <= a <= b <= c, and m has exactly three such representations}.
The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 3, m >= 0}.
EXAMPLE
a(1) = 41 = 0^2 + 4^2 + 5^2 = 1^2 + 2^2 + 6^2 = 3^3 + 4^2 + 4^2, and 41 is the first number m with A000164(m) = 3.
The representations [a,b,c] for n = 1, ..., 10, are:
n=1, 41: [0, 4, 5], [1, 2, 6], [3, 4, 4],
n=2, 50: [0, 1, 7], [0, 5, 5], [3, 4, 5],
n=3, 54: [1, 2, 7], [2, 5, 5], [3, 3, 6],
n=4, 65: [0, 1, 8], [0, 4, 7], [2, 5, 6],
n=5, 66: [1, 1, 8], [1, 4, 7], [4, 5, 5],
n=6, 74: [0, 5, 7], [1, 3, 8], [3, 4, 7],
n=7, 86: [1, 2, 9], [1, 6, 7], [5, 5, 6],
n=8, 90: [0, 3, 9], [1, 5, 8], [4, 5, 7],
n=9, 98: [0, 7, 7], [1, 4, 9], [3, 5, 8],
n=10, 99: [1, 7, 7], [3, 3, 9], [5, 5, 7].
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^2<n, 0,
`if`(b(n, i-1, t)>3, 4, min(4, b(n, i-1, t)+
`if`(i^2>n, 0, b(n-i^2, i, t-1))))))
end:
a:= proc(n) option remember; local k;
for k from 1 +`if`(n=1, 0, a(n-1))
while b(k, isqrt(k), 3)<>3 do od; k
end:
seq(a(n), n=1..100); # Alois P. Heinz, Apr 09 2013
MATHEMATICA
Select[ Range[0, 400], Length[ PowersRepresentations[#, 3, 2]] == 3 &] (* Jean-François Alcover, Apr 09 2013 *)
CROSSREFS
Cf. A000164, A005875, A000378, A094942 (one way), A224442 (two ways).
Sequence in context: A043206 A043986 A124967 * A349779 A261259 A168348
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Apr 08 2013
STATUS
approved