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A223733 Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly two ways. 5
33, 38, 41, 51, 54, 57, 59, 62, 69, 74, 77, 81, 83, 90, 94, 98, 99, 102, 105, 107, 113, 117, 118, 121, 122, 123, 125, 126, 137, 138, 139, 141, 150, 154, 155, 158, 162, 165, 170, 177, 178, 181, 187, 195, 197, 203, 210, 211, 213, 214, 217, 218, 225, 226, 229 (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)) = 2. See also A223731. These are the numbers n with exactly two 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(147) = 1885 = 16^2 + 27^2 + 30^2 = 12^2 + 29^2 + 30^2 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..147

FORMULA

This sequence lists the increasingly ordered distinct members of the set S2:= {m positive integer | m = a^2 + b^2 + c^2, 0 < a <= b <= c, and there are exactly two different solutions for this m}.

EXAMPLE

a(1) = 33 because the smallest number n with A223730(n) = 2 is 33. The two representations of 33 are denoted by  [1, 4, 4], and [2, 2, 5].

The two representations for a(n) for n = 2..10 are denoted by

n=2,  38: [1, 1, 6], [2, 3, 5],

n=3,  41: [1, 2, 6], [3, 4, 4],

n=4,  51: [1, 1, 7], [1, 5, 5],

n=4,  54: [1, 2, 7], [2, 5, 5], ([3, 3, 6] is non-primitive)

n=5,  57: [2, 2, 7], [4, 4, 5],

n=6,  59: [1, 3, 7], [3, 5, 5],

n=7,  62: [1, 5, 6], [2, 3, 7],

n=8,  69: [1, 2, 8], [2, 4, 7],

n=9,  74: [1, 3, 8], [3, 4, 7],

n=10, 77: [2, 3, 8], [4, 5, 6].

MATHEMATICA

threeSquaresCount[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ]]; Select[ Range[300], threeSquaresCount[#] == 2 &] (* Jean-Fran├žois Alcover, Jun 21 2013 *)

CROSSREFS

Cf. A223730, A223731, A223732, A223734.

Sequence in context: A180329 A020260 A140147 * A083883 A039326 A043149

Adjacent sequences:  A223730 A223731 A223732 * A223734 A223735 A223736

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Apr 05 2013

STATUS

approved

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Last modified October 23 22:42 EDT 2019. Contains 328378 sequences. (Running on oeis4.)