A007527 Numbers that are not the sum of 4 hexagonal numbers. (Formerly M3793)

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

Offset

1,1

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

Table of n, a(n) for n=1..13.

Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 143.

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

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 *)

Crossrefs

Sequence in context: A189151 A120513 A004757 * A033894 A033649 A271922

Adjacent sequences: A007524 A007525 A007526 * A007528 A007529 A007530

Keyword

fini,nonn,full

Author

N. J. A. Sloane

Status

approved