A131280 Sums of exactly 4 positive octahedral numbers A005900.

4, 9, 14, 19, 22, 24, 27, 32, 37, 40, 45, 47, 50, 52, 57, 58, 62, 63, 65, 70, 75, 76, 83, 88, 90, 93, 95, 98, 100, 101, 103, 106, 108, 111, 113, 116, 124, 126, 129, 131, 133, 136, 138, 141, 142, 149, 151, 154, 159, 164, 167, 172, 174, 176, 177, 179, 182, 185, 190

Offset: 1

Author

Jonathan Vos Post, Oct 21 2007

Keywords

• easy
• nonn

Comments

Pollock (1850) conjectured that every number is the sum of at most 7 octahedral numbers. Which octahedral numbers are themselves the sum of exactly 4 positive octahedral numbers? To begin with, Oc(3) = Oc(2) + Oc(2) + Oc(2) + Oc(1) = 6 + 6 + 6 + 1 = 19.

References

• Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005, cites the Pollock reference.
• Pollock, F. "On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders." Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.

Crossrefs

Cf. A005900, A053676-A053678, A133330.

Programs

Mathematica

With[{octs=Table[(2n^3+n)/3,{n,10}]},Take[Union[Total/@Tuples[octs,4]], 60]] (* Harvey P. Dale, Nov 26 2013 *)