A102796 -id:A102796 - cp's OEIS frontend

A104246 Minimal number of tetrahedral numbers (A000292(k) = k(k+1)(k+2)/6) needed to sum to n.

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

Offset: 1

Eric W. Weisstein, Feb 26 2005

nonn

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
Possible correction of the first comment by Sloane 2011: it appears to me from the linked reference by Salzer and Levine 1968 that 452479659 is instead the upper limit for sums of five Qx = Tx + x, where Tx are the tetrahedral numbers we want. They also mention an upper limit for sums of five Tx, which is: a(n) <= 5 for n <= 276976383. - Ewoud Dronkert, May 30 2024
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

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

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.

Cf. A000292 (tetrahedral numbers, indices of 1s), A102795 (indices of 2s), A102796 (indices of 3s), A102797 (indices of 4s), A000797 (numbers that need 5 tetrahedral numbers).
See also A102798-A102806, A102855-A102858, A193101, A193105, A281367 (the "triangular nachos" numbers).
Cf. A061336 (analog for triangular numbers).

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
# alternative
N := 10000:
L := [seq(0,i=1..N)] :
# put 1's where tetrahedral numbers reside
for i from 1 to N do
    Aj := A000292(i) ;
    if Aj <= N then
        L := subsop(Aj=1,L) ;
    end if;
end do:
for a from 1 do
    # select positions of a's, skip forward by all available Aj and
    # if that addresses a not-yet-set position in the array put a+1 there.
    for i from 1 to N do
        if op(i,L) =a then
            for j from 1 do
                Aj := A000292(j) ;
                if i+Aj <=N and op(i+Aj,L) = 0 then
                    L := subsop(i+Aj=a+1,L) ;
                end if;
                if i +Aj > N then
                    break ;
                end if;
            end do:
        end if;
    end do:
    # if all L[] are non-zero, terminate the loop
    allset := true;
    for i from 1 to N do
        if op(i,L) = 0 then
            allset := false ;
            break ;
        end if;
    end do:
    if allset then
        break ;
    end if;
end do:
seq( L[i],i=1..N) ; # R. J. Mathar, Jun 06 2025

\\ available on request. - M. F. Hasler, Mar 06 2017

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

Edited by N. J. A. Sloane, Jul 15 2011

Edited by M. F. Hasler, Mar 06 2017

A102795 Let f(n) = A104246(n) be the minimal number of nonzero tetrahedral numbers that add to n; sequence gives numbers n for which f(n) = 2.

Original entry on oeis.org

2, 5, 8, 11, 14, 21, 24, 30, 36, 39, 40, 45, 55, 57, 60, 66, 70, 76, 85, 88, 91, 94, 104, 112, 119, 121, 124, 130, 140, 155, 166, 168, 169, 175, 176, 185, 200, 204, 221, 224, 230, 240, 249, 255, 276, 285, 287, 290, 296, 304, 306, 321, 330, 340, 342

Offset: 1

Jud McCranie, Feb 26 2005

nonn

R. J. Mathar, Table of n, a(n) for n = 1..999

Cf. A000292, A104246.

See also A102796-A102806, A102855-A102858.

A000797 Numbers that are not the sum of 4 tetrahedral numbers.

Original entry on oeis.org

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

N. J. A. Sloane

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.

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).

Jud McCranie and David W. Wilson, The 241 known terms
Brady Haran and James Grime, 343867 and Tetrahedral Numbers - Numberphile, YouTube video, 2024.
Maksym Nedoshev and Viktor Kyrychenko, Peculiarities of the Operation of Complex Computational Algorithms using the Example of Testing the Polock Hypothesis [sic], Central Ukrainian Sci. Bull. (2025) Part II, Vol. 11, No. 42, 23-29. (In Ukrainian)
Jonathan Frederick 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.
Herbert E. Salzer and Norman 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

Cf. A000292 (tetrahedral numbers), A102795, A102796, A102797, A104246, A102800 (complement).

Entry revised Feb 25 2005

A343491 Number of representations of n! as a sum of 3 tetrahedral numbers (A000292).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 3, 5, 2, 3, 6, 5, 8, 8, 7, 2, 7, 8, 3, 11, 2, 2

Offset: 1

Altug Alkan, Apr 17 2021

Conjecture I: There are infinitely many n such that a(n) >= 1.
Conjecture II: Natural density of numbers n such that a(n) >= 1 is 1.
Conjecture III: Numbers n such that a(n) = 0 is a finite sequence.
Conjecture IV: a(n) >= 1 for all n.
See Links section for some solutions.

			a(4) = 2 because 4! = 0 + 4 + 20 = 4 + 10 + 10.
a(24) = 2 because 24! = f(11393630) + f(118661018) + f(127041924) = f(81298034) + f(61098204) + f(143537134) where f = A000292.

Altug Alkan, A note on sequence

Cf. A000142, A000292, A102796, A104246, A102799, A267414, A308852, A341794.

Table[Length[Solve[{i*(i + 1)*(i + 2) + j*(j + 1)*(j + 2) + k*(k + 1)*(k + 2) == 6*n!, i >= 0, j >= 0, k >= 0, i <= j, j <= k, k < (6*n!)^(1/3)}, Integers]], {n, 1, 10}] (* Vaclav Kotesovec, Apr 19 2021 *)

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A104246 Minimal number of tetrahedral numbers (A000292(k) = k(k+1)(k+2)/6) needed to sum to n.

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A102795 Let f(n) = A104246(n) be the minimal number of nonzero tetrahedral numbers that add to n; sequence gives numbers n for which f(n) = 2.

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A000797 Numbers that are not the sum of 4 tetrahedral numbers.

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A343491 Number of representations of n! as a sum of 3 tetrahedral numbers (A000292).

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