A007527 Numbers that are not the sum of 4 hexagonal numbers.

5, 10, 11, 20, 25, 26, 38, 39, 54, 65, 70, 114, 130

Offset: 1

Author

N. J. A. Sloane

Keywords

• fini
• nonn
• full

Comments

The sequence is complete. "In 1830, Legendre (1979) proved that every number larger than 1791 is a sum of four hexagonal numbers". See Eric Weisstein's link and Legendre reference. It is easy to check all numbers <= 1791 by computer. - Olivier Pirson, Sep 14 2007

References

• A.-M. Legendre, Theorie des nombres, 4th ed., 2 vols. Paris: A. Blanchard, 1979.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 143.
• Shyam Sunder Gupta, Unlucky 13, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 13, 353-365.
• R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.
• Eric Weisstein's World of Mathematics, Hexagonal Number

Programs

Mathematica

lim = 1791; maxa = Ceiling[a /. Last[Solve[a(2a - 1) == lim]]]; t = Flatten[ Table[a(2a - 1) + b(2b - 1) + c(2c - 1) + d(2d - 1), {a, 0, maxa}, {b, 0, a}, {c, 0, b}, {d, 0, c}], 3]; Complement[ Range[lim], t](* Jean-François Alcover, Sep 21 2011 *)