A000797 Numbers that are not the sum of 4 tetrahedral numbers. (Formerly M5033 N2172)

17, 27, 33, 52, 73, 82, 83, 103, 107, 137, 153, 162, 217, 219, 227, 237, 247, 258, 268, 271, 282, 283, 302, 303, 313, 358, 383, 432, 437, 443, 447, 502, 548, 557, 558, 647, 662, 667, 709, 713, 718, 722, 842, 863, 898, 953, 1007, 1117, 1118

Offset

1,1

Comments

It is an open problem of long standing ("Pollock's Conjecture") to show that this sequence is finite.

More precisely, Salzer and Levine conjecture that every number is the sum of at most 5 tetrahedral numbers and in fact that there are exactly 241 numbers (the terms of this sequence) that require 5 tetrahedral numbers, the largest of which is 343867.

References

L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 22.

S. S. Skiena, The Algorithm Design Manual, Springer-Verlag, 1998, pp. 43-45 and 135-136.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Jud McCranie and David W. Wilson, The 241 known terms

B. Haran and J. Grime, 343867 and Tetrahedral Numbers - Numberphile, YouTube video, 2024.

F. Pollock, 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, Proc. Roy. Soc. London, 5 (1851), 922-924.

H. E. Salzer and N. Levine, Table of integers not exceeding 10 00000 that are not expressible as the sum of four tetrahedral numbers, Math. Comp., 12 (1958), 141-144.

Eric Weisstein's World of Mathematics, Pollock's Conjecture

Eric Weisstein's World of Mathematics, Tetrahedral Number

Crossrefs

Cf. A000292 (tetrahedral numbers), A102795, A102796, A102797, A104246.

Sequence in context: A268330 A221282 A033702 * A171168 A147202 A146776

Adjacent sequences: A000794 A000795 A000796 * A000798 A000799 A000800

Keyword

nonn,fini

Author

N. J. A. Sloane

Extensions

Entry revised Feb 25 2005

Status

approved