A104246 -id:A104246 - cp's OEIS frontend

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.

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.

A102796 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) = 3.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 22, 25, 28, 31, 34, 37, 41, 43, 44, 46, 49, 50, 58, 59, 61, 64, 65, 67, 71, 74, 75, 77, 80, 86, 89, 90, 92, 95, 96, 98, 101, 105, 108, 111, 113, 114, 116, 122, 123, 125, 126, 128, 129, 131, 132, 134, 139, 141, 144, 147, 150, 154, 156

Offset: 1

Jud McCranie, Feb 26 2005

nonn

R. J. Mathar, Table of n, a(n) for n = 1..4356

Cf. A000292, A104246, A102795, etc.

A102797 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) = 4.

Original entry on oeis.org

7, 13, 16, 19, 23, 26, 29, 32, 38, 42, 47, 48, 51, 53, 54, 62, 63, 68, 69, 72, 78, 79, 81, 87, 93, 97, 99, 100, 102, 106, 109, 110, 115, 117, 118, 127, 133, 135, 136, 138, 142, 143, 145, 146, 148, 149, 151, 152, 157, 158, 161, 163, 164, 171, 174, 181

Offset: 1

Jud McCranie, Feb 26 2005

nonn

R. J. Mathar, Table of n, a(n) for n = 1..4836

Cf. A000292, A104246, A102795, etc.

A102798 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

1, 2, 4, 5, 8, 10, 11, 14, 20, 21, 24, 30, 35, 36, 39, 40, 45, 55, 56, 57, 60, 66, 70, 76, 84, 85, 88, 91, 94, 104, 112, 119, 120, 121, 124, 130, 140, 155, 165, 166, 168, 169, 175, 176, 185, 200, 204, 220, 221, 224, 230, 240, 249, 255, 276, 285, 286, 287

Offset: 1

Jud McCranie, Feb 26 2005

nonn

R. J. Mathar, Table of n, a(n) for n = 1..1047

Cf. A000292, A104246, A102795, etc.

A102800 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) <= 4.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74

Offset: 1

Jud McCranie, Feb 26 2005

nonn

Cf. A000292, A000797, A104246, A102795, etc.

A102799 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) <= 3.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 18, 20, 21, 22, 24, 25, 28, 30, 31, 34, 35, 36, 37, 39, 40, 41, 43, 44, 45, 46, 49, 50, 55, 56, 57, 58, 59, 60, 61, 64, 65, 66, 67, 70, 71, 74, 75, 76, 77, 80, 84, 85, 86, 88, 89, 90, 91, 92, 94, 95, 96, 98, 101, 104, 105

Offset: 1

Jud McCranie, Feb 26 2005

nonn

Cf. A000292, A104246, A102795, etc.

A061336 Smallest number of triangular numbers which sum to n.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 1, 2, 3, 2, 1, 2, 2, 2, 3, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 3, 2, 3, 1, 2, 2, 2, 3, 3, 2, 2, 3, 1, 2, 3, 2, 2, 3, 2, 3, 3, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 1, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3

Offset: 0

Henry Bottomley, Apr 25 2001

nonn

a(n)=3 if n=5 or 8 mod 9, since triangular numbers are {0,1,3,6} mod 9.
From Bernard Schott, Jul 16 2022: (Start)
In September 1636, Fermat, in a letter to Mersenne, made the statement that every number is a sum of at most three triangular numbers. This was proved by Gauss, who noted this event in his diary on July 10 1796 with the notation:
EYPHKA! num = DELTA + DELTA + DELTA (where Y is in fact the Greek letter Upsilon and DELTA is the Greek letter of that name).
This proof was published in his book Disquisitiones Arithmeticae, Leipzig, 1801. (End)

			a(3)=1 since 3=3, a(4)=2 since 4=1+3, a(5)=3 since 5=1+1+3, with 1 and 3 being triangular.

Elena Deza and Michel Marie Deza, Fermat's polygonal number theorem, Figurate numbers, World Scientific Publishing (2012), Chapter 5, pp. 313-377.
C. F. Gauss, Disquisitiones Arithmeticae, Yale University Press, 1966, New Haven and London, p. 342, art. 293.

Giovanni Resta, Table of n, a(n) for n = 0..10000
George E. Andrews, EYPHKA! num = Delta + Delta + Delta, J. Number Theory 23 (1986), 285-293.
Eric Weisstein's World of Mathematics, Fermat's Polygonal Number Theorem.

Cf. A000217, A007294, A057945, A061337, A051533, A020757.

Cf. A100878 (analog for A000326), A104246 (analog for A000292), A283365 (analog for A000332), A283370 (analog for A000389).

t[n_]:=n*(n+1)/2; a[0]=0; a[n_]:=Block[ {k=1, tt= t/@ Range[Sqrt[2*n]]}, Off[IntegerPartitions::take]; While[{} == IntegerPartitions[n, {k}, tt, 1], k++]; k]; a/@ Range[0, 104] (* Giovanni Resta, Jun 09 2015 *)

\\ see A283370 for generic code, working but not optimized for this case of triangular numbers. - M. F. Hasler, Mar 06 2017

a(n)=my(m=n%9,f); if(m==5 || m==8, return(3)); f=factor(4*n+1); for(i=1,#f~, if(f[i,2]%2 && f[i,1]%4==3, return(3))); if(ispolygonal(n,3), n>0, 2) \\ Charles R Greathouse IV, Mar 17 2022

a(n) = 0 if n=0, otherwise 1 if n is in A000217, otherwise 2 if n is in A051533, otherwise 3 in which case n is in A020757.

a(n) <= 3 (proposed by Fermat and proved by Gauss). - Bernard Schott, Jul 16 2022

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

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

A102806 Numbers that are not the sum of distinct tetrahedral numbers.

Original entry on oeis.org

2, 3, 6, 7, 8, 9, 12, 13, 16, 17, 18, 19, 22, 23, 26, 27, 28, 29, 32, 33, 37, 38, 41, 42, 43, 44, 47, 48, 51, 52, 53, 54, 58, 62, 63, 64, 68, 72, 73, 74, 75, 78, 79, 82, 83, 93, 97, 100, 103, 107, 110, 113, 117, 127, 128, 132, 136, 137, 138, 142, 146, 147, 148

Offset: 1

Jud McCranie, Feb 26 2005

The b-file contains all the members of the sequence. See link. - Robert Israel, Dec 29 2019

Robert Israel, Table of n, a(n) for n = 1..112
Robert Israel, Proof that the b-file is full

Cf. A000292, A104246, A102795, etc.

N:= 100000: # to test all n <= 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 false fi;
   for s in tets while s < min(n, tmax) do
     if not procname(n-s,s) then return false fi
   od;
     true
end proc:
select(f, [$1..N],infinity); # Robert Israel, Dec 29 2019

M = 1000; (* to test all n <= M *)
ft[t_] := t*(t+1)*(t+2)/6;
tets = Map[ft, Range[Floor[(6*M)^(1/3)]]];
f[n_, tMax_] := f[n, tMax] = Module[{res, s}, If[MemberQ[tets, n] && n < tMax, Return[False]]; For[i = 1, s = tets[[i]]; i <= Length[tets] && s < Min[n, tMax], i++, If[!f[n-s, s], Return[False]]]; True];
Select[Range[M], f[#, Infinity]&] (* Jean-François Alcover, Sep 15 2022, after Robert Israel *)

cp's OEIS Frontend

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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A102796 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) = 3.

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A102797 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) = 4.

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A102798 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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A102800 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) <= 4.

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A102799 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) <= 3.

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A061336 Smallest number of triangular numbers which sum to n.

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

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

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A102806 Numbers that are not the sum of distinct tetrahedral numbers.

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