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A223734 Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly three ways. 5
66, 86, 89, 101, 110, 114, 131, 149, 153, 166, 171, 173, 174, 179, 182, 185, 186, 189, 198, 219, 221, 222, 227, 233, 234, 237, 241, 242, 245, 258, 261, 270, 274, 286, 291, 294, 302, 305, 309, 318, 323, 334, 338, 347, 349, 361, 363, 366, 377, 378, 387, 405, 410 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

These are the increasingly ordered numbers a(n) for which A223730(a(n)) = 3. See also A223731. These are the numbers n with exactly three representation as a primitive sum of three nonzero squares (not taking into account the order of the three terms, and the number to be squared for each term is taken positive).

Conjecture: a(185) = 4075 = 31^2 + 33^2 + 45^2 = 23^2 + 39^2 + 45^2 = 5^2 + 9^2 + 63^2 is the largest element of this sequence. - Alois P. Heinz, Apr 06 2013

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..185

FORMULA

This sequence lists the increasingly ordered distinct members of the set S3:= {m positive integer | m = a^2 + b^2 + c^2, 0 < a <= b <= c, and there are exactly three different solutions for this m}.

EXAMPLE

a(1) = 66 because the smallest number n with A223730(n) = 3 is 66. The three solutions for m = 66 are denoted by [1,1,8], [1,4,7] and [4,5,5].

For n=2..10 the three representations of a(n) are given by

n=2,   86:  [1, 2, 9],  [1, 6, 7], [5, 5, 6],

n=3,   89:  [2, 2, 9],  [2, 6, 7], [3, 4, 8],

n=4,  101:  [1, 6, 8],  [2, 4, 9], [4, 6, 7],

n=5,  110:  [1, 3, 10], [2, 5, 9], [5, 6, 7],

n=6,  114:  [1, 7, 8],  [4, 7, 7], [5, 5, 8],

n=7,  131:  [1, 3, 11], [1, 7, 9], [5, 5, 9],

n=8,  149:  [1, 2, 12], [2, 8, 9], [6, 7, 8],

n=9,  153:  [2, 7, 10], [4, 4, 11], [5, 8, 8],

n=10: 166:  [2, 9, 9],  [3, 6, 11], [6, 7, 9].

For n = 153 there is also the non-primitive representation [6,6,9] = 3*[2,2,3] not counted here.

MATHEMATICA

threeSquaresCount[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ]]; Select[ Range[500], threeSquaresCount[#] == 3 &] (* Jean-Fran├žois Alcover, Jun 21 2013 *)

CROSSREFS

Cf. A223730, A223731, A223732, A223733.

Sequence in context: A293175 A032485 A257029 * A031411 A098775 A109759

Adjacent sequences:  A223731 A223732 A223733 * A223735 A223736 A223737

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Apr 05 2013

STATUS

approved

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Last modified October 23 23:05 EDT 2019. Contains 328379 sequences. (Running on oeis4.)