

A104246


Minimal number of tetrahedral numbers (A000292(k) = k(k+1)(k+2)/6) needed to sum to n.


25



1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 2, 3, 4, 3, 3, 2, 3, 4, 4, 3, 3, 4, 5, 4, 4, 2, 1, 2, 3, 3, 2, 3, 4, 4, 3, 3, 2, 3, 4, 4, 2, 3, 4, 5, 3, 3, 2, 3, 4, 4, 3, 4, 5, 5, 1, 2, 3, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 3, 4, 4, 3, 4
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OFFSET

1,2


COMMENTS

According to Dickson, Pollock conjectures that a(n) <= 5 for all n. Watson shows that a(n) <= 8 for all n, and Salzer and Levine show that a(n) <= 5 for n <= 452479659.  N. J. A. Sloane, Jul 15 2011
If we use the greedy algorithm for this, we get A281367.  N. J. A. Sloane, Jan 30 2017
Could be extended with a(0) = 0, in analogy to A061336. Kim (2003, p.73) shows that a(n) <= 5 for all n.  M. F. Hasler, Mar 06 2017


REFERENCES

Dickson, L. E., History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 1952, see p. 13.


LINKS

Lars Blomberg, Table of n, a(n) for n = 1..10000
Hyun Kwang Kim, On regular polytope numbers, Proc. Amer. Math. Soc. 131 (2003), p. 6575.
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, Abs. Papers Commun. Roy. Soc. London 5, 922924, 18431850.
H. E. Salzer and N. Levine, Table of Integers Not Exceeding 1000000 that are Not Expressible as the Sum of Four Tetrahedral Numbers, Math. Comp. 12, 141144, 1958.
H. E. Salzer and N. Levine, Proof that every integer <= 452,479,659 is a sum of five numbers of the form Q_x = (x^3+5x)/6, x>= 0, Math. Comp., (1968), 191192.
N. J. A. Sloane, Transforms
G. L. Watson, Sums of eight values of a cubic polynomial, J. London Math. Soc., 27 (1952), 217224.
Eric Weisstein's World of Mathematics, Tetrahedral Number


MAPLE

tet:=[seq((n^3n)/6, n=1..20)];
LAGRANGE(tet, 8, 120); # the LAGRANGE transform of a sequence is defined in A193101.  N. J. A. Sloane, Jul 15 2011


PROG

(PARI) available on request.  M. F. Hasler, Mar 06 2017
(PARI)
seq(N) = {
my(a = vector(N, k, 8), T = k>(k*(k+1)*(k+2))\6);
for (n = 1, N,
my (k1 = sqrtnint((6*n)\8, 3), k2 = sqrtnint(6*n, 3));
while(n < T(k2), k2); if (n == T(k2), a[n] = 1; next());
for (k = k1, k2, a[n] = min(a[n], a[n  T(k)] + 1))); a;
};
seq(102) \\ Gheorghe Coserea, Mar 14 2017


CROSSREFS

Cf. A000292 (tetrahedral numbers), A000797 (numbers that need 5 tetrahedral numbers).
See also A102795A102806, A102855A102858, A193101, A193105, A281367 (the "triangular nachos" numbers).
Cf. A061336 (analog for triangular numbers).
Sequence in context: A096436 A053610 A264031 * A281367 A007720 A129968
Adjacent sequences: A104243 A104244 A104245 * A104247 A104248 A104249


KEYWORD

nonn


AUTHOR

Eric W. Weisstein, Feb 26 2005


EXTENSIONS

Edited by N. J. A. Sloane, Jul 15 2011
Edited by M. F. Hasler, Mar 06 2017


STATUS

approved



