

A223732


Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly one way.


5



3, 6, 9, 11, 14, 17, 18, 19, 21, 22, 26, 27, 29, 30, 34, 35, 42, 43, 45, 46, 49, 50, 53, 61, 65, 67, 70, 73, 75, 78, 82, 91, 93, 97, 106, 109, 115, 133, 142, 145, 147, 157, 163, 169, 190, 193, 202, 205, 235, 253, 265, 277, 298, 397, 403, 427, 442, 445, 505, 793
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OFFSET

1,1


COMMENTS

These are the increasingly ordered numbers a(n) for which A233730(a(n)) = 1. See also A233731. These are the numbers n with exactly one 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: 793 = 6^2 + 9^2 + 26^2 is the largest element of this sequence.  Alois P. Heinz, Apr 06 2013


LINKS

Table of n, a(n) for n=1..60.
Eugen J. Ionascu, Ehrhart polynomial for lattice squares, cubes and hypercubes, arXiv:1508.03643 [math.NT], 2015.


FORMULA

This sequence lists the increasingly ordered members of the set S1 := {m positive integer  m = a^2 + b^2 + c^2, 0 < a <= b <= c, gcd(a,b,c) = 1, with only one such solution for this m}.


EXAMPLE

a(1) = 3 because there is no solution for m = 1 and 2 as a primitive sum of three nonzero squares, and m = 3 = 1^2 + 1^2 + 1^2 is the only solution with [a,b,c] = [1,1,1].
a(5) = 14 because 14 is the fifth largest member of the set S1, and [a,b,c] = [1,2,3] denotes this unique representation for m = 14.


MATHEMATICA

threeSquaresCount[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ]]; Select[ Range[800], threeSquaresCount[#] == 1 &] (* JeanFrançois Alcover, Jun 21 2013 *)


CROSSREFS

Cf. A233730, A233731, A233733, A233734.
Sequence in context: A310144 A310145 A223731 * A094740 A305849 A047400
Adjacent sequences: A223729 A223730 A223731 * A223733 A223734 A223735


KEYWORD

nonn


AUTHOR

Wolfdieter Lang, Apr 05 2013


STATUS

approved



