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A224443 Numbers that are the sum of three squares (square 0 allowed) in exactly three ways. 10
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 (list; graph; refs; listen; history; text; internal format)
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

Alois P. Heinz, Table of n, a(n) for n = 1..1000

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-Fran├žois Alcover, Apr 09 2013 *)

CROSSREFS

Cf. A000164, A005875, A000378, A094942 (one way), A224442 (two ways).

Sequence in context: A043206 A043986 A124967 * A261259 A168348 A052032

Adjacent sequences:  A224440 A224441 A224442 * A224444 A224445 A224446

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Apr 08 2013

STATUS

approved

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Last modified November 28 13:47 EST 2021. Contains 349413 sequences. (Running on oeis4.)