

A223735


Positive numbers that are not representable as a primitive sum of three nonzero squares.


2



1, 2, 4, 5, 7, 8, 10, 12, 13, 15, 16, 20, 23, 24, 25, 28, 31, 32, 36, 37, 39, 40, 44, 47, 48, 52, 55, 56, 58, 60, 63, 64, 68, 71, 72, 76, 79, 80, 84, 85, 87, 88, 92, 95, 96, 100, 103, 104, 108, 111, 112, 116, 119, 120, 124, 127, 128, 130, 132, 135, 136, 140, 143, 144
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OFFSET

1,2


COMMENTS

This is the complement of A223731. There an F. HalterKoch reference is given.


LINKS



FORMULA

a(n) has no representation as a^2 + b^2 + c^2 with 0 < a <= b <= c and gcd(a,b,c) = 1.
Conjectured g.f.: (2*x^61 x^60 +2*x^59 x^58 2*x^57 +x^43 +3*x^42 3*x^41 +x^40 2*x^39 +2*x^32 x^31 +2*x^30 x^29 2*x^28 +x^23 +3*x^22 3*x^21 +x^20 2*x^19 +x^18 +2*x^16 3*x^14 +x^12 +3*x^11 x^10 +x^6 x^5 +x^4 +2*x^2 +x +1)*x / (x^4 x^3 x +1).  Alois P. Heinz, Apr 06 2013


EXAMPLE

For a(1) up to a(7) there is no representation as sum of three nonzero squares.
a(8) = 12 because the only representation of 12 as a sum of nonzero squares is given by [a,b,c] = [2,2,2] = 2*[1,1,1], and this is not a primitive sum because gcd(2,2,2) = 2, not 1.


MATHEMATICA

notThreeSquaresQ[n_] := Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ] == {}; Select[Range[200], notThreeSquaresQ] (* JeanFrançois Alcover, Jun 21 2013 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



