A117089 Primes that are not the sum of 3 hexagonal numbers.

5, 11, 19, 23, 37, 41, 53, 59, 83, 89, 113, 131, 167, 173, 179, 229, 251, 269, 293, 313, 317, 383, 389, 439, 443, 509, 599, 641, 683, 859, 929, 1031, 1033, 1049, 1163, 1193, 1283, 1301, 1303, 1307, 1439, 1493, 1499, 1543, 1619, 1733, 2143, 2153, 2333, 2687, 2693, 3083, 3089, 3533, 3719, 3989, 4003, 4583, 4673, 4703, 5387, 5651, 5849, 5903, 6173, 6389, 6449, 7481, 9293, 12113, 15803, 16433, 19763, 61403

Offset: 1

Author

Jonathan Vos Post, Apr 18 2006

Keywords

• easy
• fini
• nonn

Examples

			5 is the sum of five hexagonal numbers; 11 is the sum of six hexagonal numbers; the other 72 primes are the sum of four hexagonal numbers. - _T. D. Noe_, Apr 20 2006
		

References

• Legendre, Théorie des Nombres, 3rd edition, 1830.

Links

• W. Duke and R. Schulze-Pilot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99(1990), 49-57.
• R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.

Crossrefs

Cf. A000040, A000384, A007527, A007536, A117065.

Programs

Mathematica

nn=201; hex=Table[n(2n-1), {n,0,nn-1}]; ps=Prime[Range[PrimePi[hex[[ -1]]]]]; Do[n=hex[[i]]+hex[[j]]+hex[[k]]; If[n<=hex[[ -1]]&&PrimeQ[n], ps=DeleteCases[ps,n]], {i,nn}, {j,i,nn}, {k,j,nn}]; ps (* T. D. Noe, Apr 20 2006 *)

Formula

A000040 INTERSECT A007536.

Extensions

More terms from T. D. Noe, who conjectures that the list shown here is complete. His search up to 7*10^7 gave no further terms. - Apr 20 2006