A102806 -id:A102806 - 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.

A102857 Sum of 1, 2 or 3 distinct tetrahedral numbers.

Original entry on oeis.org

0, 1, 4, 5, 10, 11, 14, 15, 20, 21, 24, 25, 30, 31, 34, 35, 36, 39, 40, 45, 46, 49, 55, 56, 57, 59, 60, 61, 65, 66, 67, 70, 76, 77, 80, 84, 85, 86, 88, 89, 91, 92, 94, 95, 98, 101, 104, 105, 108, 111, 114, 119, 120, 121, 123, 124, 125, 129, 130, 131, 134, 139

Offset: 1

Jud McCranie, Mar 01 2005

nonn

R. J. Mathar, Table of n, a(n) for n = 1..4545

Cf. A104246, A102795-A102801, A102803-A102806, A102855, A102857, A102858.

Cf. A102856 (subsequence), A102802 (subsequence 3 dist. posit.).

Module[{nn=10,tetra},tetra=Table[(n(n+1)(n+2))/6,{n,nn}];Rest[Union[ Total/@ Subsets[tetra,3]]]] (* Harvey P. Dale, Nov 17 2017 *)

a(1)=0 prepended. - R. J. Mathar, Jun 05 2025

A102855 Minimal number of distinct nonzero tetrahedral numbers needed to represent n, or -1 if no such representation is possible.

Original entry on oeis.org

1, -1, -1, 1, 2, -1, -1, -1, -1, 1, 2, -1, -1, 2, 3, -1, -1, -1, -1, 1, 2, -1, -1, 2, 3, -1, -1, -1, -1, 2, 3, -1, -1, 3, 1, 2, -1, -1, 2, 3, -1, -1, -1, -1, 2, 3, -1, -1, 3, 4, -1, -1, -1, -1, 2, 1, 2, -1, 3, 2, 3, -1, -1, -1, 3, 2, 3, -1, 4, 3, 4, -1, -1, -1, -1, 2, 3, -1, -1

Offset: 1

Jud McCranie, Mar 01 2005

sign

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A104246, A102795-A102806, A102856-A102858.

N:= 100; # for a(1)..a(N)
ft:= t -> t*(t+1)*(t+2)/6:
tets:= map(ft, [$1..floor((6*N)^(1/3))]:
f:= proc(n,tmax) option remember;
   local res, s;
   if member(n, tets) and n < tmax then return 1 fi;
   min(seq(1 + procname(n-s,s), s=select(`<`,tets,min(n,tmax))));
end proc:
subs(infinity=-1,map(f, [$1..N],infinity)); # Robert Israel, Dec 29 2019

M = 100; (* number of terms *)
ft[t_] := t(t+1)(t+2)/6;
tets = ft /@ Range[1, Floor[(6M)^(1/3)]];
f[n_, tmax_] := f[n, tmax] = If[MemberQ[tets, n] && n < tmax, 1, Min[ Table[1 + f[n-s, s], {s, Select[tets, # < Min[n, tmax]&]}]]];
f[#, Infinity]& /@ Range[1, M] /. Infinity -> -1 (* Jean-François Alcover, Aug 05 2022, after Robert Israel *)

A102856 Sum of 1 or 2 distinct tetrahedral numbers.

Original entry on oeis.org

0, 1, 4, 5, 10, 11, 14, 20, 21, 24, 30, 35, 36, 39, 45, 55, 56, 57, 60, 66, 76, 84, 85, 88, 91, 94, 104, 119, 120, 121, 124, 130, 140, 155, 165, 166, 169, 175, 176, 185, 200, 204, 220, 221, 224, 230, 240, 249, 255, 276, 285, 286, 287, 290, 296, 304, 306

Offset: 1

Jud McCranie, Mar 01 2005

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000292 (subsequence), A010329, A104246, A102795-A102800, A102802-A102806, A102855, A102857, A102858, A334013.

Cf. A102801 (sums if 2 distinct positive)

N:= 1000: # for terms <= N
R:= 0:
for i from 1 do
  ti:= i*(i+1)*(i+2)/6;
  if ti > N then break fi;
  for j from 0 to i-1 do
     v:= ti + j*(j+1)*(j+2)/6;
     if v > N then break fi;
     R:= R, v;
  od;
od:
sort(convert({R},list)); # Robert Israel, Jan 16 2024

Module[{nn=12, tetra}, tetra=Table[(n(n+1)(n+2))/6, {n, nn}]; Union[ Total/@ Subsets[tetra, 2]]] (* James C. McMahon, Jan 12 2024 *)

a(1) = 0 prepended by James C. McMahon, Jan 15 2024

cp's OEIS Frontend

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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A102857 Sum of 1, 2 or 3 distinct tetrahedral numbers.

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A102855 Minimal number of distinct nonzero tetrahedral numbers needed to represent n, or -1 if no such representation is possible.

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A102856 Sum of 1 or 2 distinct tetrahedral numbers.

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A102858 Sum of 1, 2, or 4 distinct tetrahedral numbers.

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A352350 a(n) is the largest number that is not the sum of distinct n-gonal pyramidal numbers.

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