A223735 Positive numbers that are not representable as a primitive sum of three nonzero squares.

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

Offset

1,2

Comments

This is the complement of A223731. There an F. Halter-Koch reference is given.

Links

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

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] (* Jean-François Alcover, Jun 21 2013 *)

Crossrefs

Cf. A223730, A223731, A223732, A223733, A223734.

Sequence in context: A195176 A195126 A047496 * A123663 A174131 A014248

Adjacent sequences: A223732 A223733 A223734 * A223736 A223737 A223738

Keyword

nonn

Author

Wolfdieter Lang, Apr 06 2013

Status

approved