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 A104246 Minimal number of tetrahedral numbers (A000292(k) = k(k+1)(k+2)/6) needed to sum to n. 35
 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 (list; graph; refs; listen; history; text; internal format)
 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, first row of table "d = 3" on p. 73) gives max {a(n)} = 5 as a "numerical result", but the value has no "* denoting exact values" (see Remark at end of paper), which means this could be incorrect. - M. F. Hasler, Mar 06 2017, edited Sep 22 2022 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), pp. 65-75. 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, 922-924, 1843-1850. 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, 141-144, 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), 191-192. N. J. A. Sloane, Transforms G. L. Watson, Sums of eight values of a cubic polynomial, J. London Math. Soc., 27 (1952), 217-224. Eric Weisstein's World of Mathematics, Tetrahedral Number MAPLE tet:=[seq((n^3-n)/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 A102795-A102806, A102855-A102858, A193101, A193105, A281367 (the "triangular nachos" numbers). Cf. A061336 (analog for triangular numbers). Sequence in context: A053610 A264031 A338482 * 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

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