A060773 -id:A060773 - cp's OEIS frontend

A053614 Numbers that are not the sum of distinct triangular numbers.

2, 5, 8, 12, 23, 33

Offset: 1

Jud McCranie, Mar 19 2000

The Mathematica program first computes A024940, the number of partitions of n into distinct triangular numbers. Then it finds those n having zero such partitions. It appears that A024940 grows exponentially, which would preclude additional terms in this sequence. - T. D. Noe, Jul 24 2006, Jan 05 2009

			a(2) = 5: the 7 partitions of 5 are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. Among those the distinct ones are 5, 4+1, 3+2. None contains all distinct triangular numbers.
12 is a term as it is not a sum of 1, 3, 6 or 10 taken at most once.

Joe Roberts, Lure of the Integers, The Mathematical Association of America, 1992, page 184, entry 33.
David Wells in "The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, page 94, states that "33 is the largest number that is not the sum of distinct triangular numbers".

Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.

Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A001422, A121405 (corresponding sequences for square and pentagonal numbers)
Cf. A000217, A002243, A002244, A014134, A014156, A014158, A020757, A050941, A050942, A051611, A007294, A051533, A060773.

nn=100; t=Rest[CoefficientList[Series[Product[(1+x^(k*(k+1)/2)), {k,nn}], {x,0,nn(nn+1)/2}], x]]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)

Complement of A061208.

Entry revised by N. J. A. Sloane, Jul 23 2006

A061262 Smallest number representable as the sum of 3 triangular numbers in exactly n ways.

Original entry on oeis.org

0, 3, 12, 21, 52, 57, 91, 121, 136, 211, 192, 226, 409, 331, 367, 406, 511, 507, 886, 637, 772, 721, 871, 952, 1102, 1066, 1227, 1192, 1641, 1621, 1396, 1381, 1501, 1732, 1792, 1927, 1942, 2401, 2611, 2551, 2422, 2557, 2887, 2821, 3136, 3271, 3607, 3376

Offset: 1

Ed Pegg Jr, Apr 24 2001

Fermat claimed, Euler tried, Gauss proved (July 10, 1796) that every number can be represented as a sum of three triangular numbers. I'm considering 0 as a triangular number here. If at first you do not succeed, tri + tri + tri again.
Conjecture: for n large enough, 1 < a(n)/n^2 < 2. - Benoit Cloitre, May 10 2003
Conjecture: No term a(n) with n > 2 is congruent to 0 or 3 modulo 5. - Zhi-Wei Sun, Feb 25 2015

			57 is the smallest number that can be represented by exactly 6 different triangular triple sums: {6, 6, 5}, {7, 7, 1}, {8, 5, 3}, {8, 6, 0}, {9, 3, 3}, {10, 1, 1}.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000217, A053614, A060773, A002636.

Cf. A124978.

a = Table[ n(n + 1)/2, {n, 0, 85} ]; b = {0}; c = Table[0, {3655} ]; Do[ b = Append[b, a[[i] ] + a[[j]] + a[[k]]], {k, 1, 85}, {j, 1, k}, {i, 1, j} ]; b = Delete[b, 1]; b = Sort[b]; l = Length[b]; Do[ If[b[[n]] < 3655, c[[b[[n]] + 1]]++ ], {n, 1, l} ]; Do[ k = 1; While[ c[[k]] != n, k++ ]; Print[k - 1], {n, 1, 48} ]

A274046 a(n) is the smallest positive integer which can be represented as the sum of distinct positive triangular numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 10, 25, 31, 49, 46, 55, 67, 70, 76, 82, 117, 102, 91, 97, 107, 101, 135, 110, 112, 116, 115, 119, 128, 0, 131, 133, 130, 148, 145, 136, 0, 137, 149, 154, 146, 0, 169, 152, 157, 155, 168, 171, 158, 174, 161, 0, 183, 184, 167, 0, 0, 173, 0, 175, 181, 190

Offset: 1

Phil Scovis, Jun 07 2016

nonn

46 is the smallest number that can be expressed as the sum of distinct triangular numbers in five ways, but 49 is the smallest that can be so expressed in exactly five ways. There are further examples of this phenomenon.

			25 = 1 + 3 + 6 + 15 = 10 + 15 = 1 + 3 + 21. This is the smallest number that can be written as the sum of distinct triangular numbers in three different ways. So a(3)=25.
The first null values of a(n) occur for n = 25, 32, 37, 47, 51, 52, 54, 61,... - _Giovanni Resta_, Jun 08 2016

Cf. A007294, A060773, A024940, A064816.

nT[n_, m_: 0] := nT[n,m] = If[n == 0, 1, Block[{t, i=m+1, s=0}, While[(t = i*(i+1)/2) <= n, s += nT[n-t, i]; i++]; s]]; a[n_] := Block[{k=0, t}, While[(t = nT[++k]) != n && t < Max[2*n, 30]]; If[t == n, k, 0]]; Array[a, 57] (* Giovanni Resta, Jun 08 2016 *)

a(15)-a(20) from Tom Edgar, Jun 08 2016

a(21)-a(57) from Giovanni Resta, Jun 08 2016

A117048 Prime numbers that are expressible as the sum of two positive triangular numbers.

Original entry on oeis.org

2, 7, 11, 13, 29, 31, 37, 43, 61, 67, 73, 79, 83, 97, 101, 127, 137, 139, 151, 157, 163, 181, 191, 193, 199, 211, 227, 241, 263, 277, 281, 307, 331, 353, 367, 373, 379, 389, 409, 421, 433, 443, 461, 463, 487, 499, 541, 571, 577, 587, 601, 619, 631, 659, 661

Offset: 1

Andrew S. Plewe, Apr 15 2006

If the triangular number 0 is allowed, only one additional prime occurs: 3. In that case, the sequence becomes A117112.

A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

			2 = 1 + 1
7 = 1 + 6
11 = 1 + 10
13 = 10 + 3, etc.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000040, A000217, A002243, A002244, A020756, A051533, A053614, A060773, A002636.

tri = Table[n (n + 1)/2, {n, 40}]; Select[Union[Flatten[Outer[Plus, tri, tri]]], # <= tri[[-1]]+1 && PrimeQ[#] &] (* T. D. Noe, Apr 07 2011 *)

is(n)=for(k=sqrtint(4*n+1)\2+1,(sqrtint(8*n+1)-1)\2, if(ispolygonal(n-k*(k+1)/2,3), return(n>3 && isprime(n)))); n==2 \\ Charles R Greathouse IV, Nov 07 2014

A111638 Numbers having a unique partition into three positive triangular numbers.

Original entry on oeis.org

3, 5, 7, 8, 9, 10, 13, 14, 15, 16, 18, 24, 25, 36, 38, 50, 53, 55, 60, 69, 81, 83, 99, 110, 119

Offset: 1

T. D. Noe, Aug 10 2005

A063993(a(n)) = 1. - Reinhard Zumkeller, Jul 20 2012

			Example: 119=55+36+28

Cf. A060773 (n having a unique partition into three nonnegative triangular numbers).

Cf. A002097, A064825.

trig[n_]:=n(n+1)/2; trigInv[x_]:=Ceiling[Sqrt[Max[0, 2x]]]; lim=100; nLst=Table[0, {trig[lim]}]; Do[n=trig[a]+trig[b]+trig[c]; If[n>0 && n<=trig[lim], nLst[[n]]++ ], {a, 1, lim}, {b, a, trigInv[trig[lim]-trig[a]]}, {c, b, trigInv[trig[lim]-trig[a]-trig[b]]}]; Flatten[Position[nLst, 1]]

A071530 Numbers that are the sum of 3 triangular numbers in exactly 2 ways.

Original entry on oeis.org

3, 6, 7, 9, 10, 13, 15, 17, 18, 19, 23, 24, 25, 26, 32, 33, 35, 38, 41, 44, 47, 54, 60, 62, 68, 69, 74, 80, 83, 89, 95, 99, 110, 113, 119, 128, 179, 194

Offset: 1

Benoit Cloitre, Jun 02 2002

nonn

If it is required that the triangular numbers be positive, sequence A064825 results. - Jon E. Schoenfield, Jan 01 2020

			From _Jon E. Schoenfield_, Jan 01 2020: (Start)
15 is a term of the sequence because there are exactly 2 ways to express 15 as the sum of 3 triangular numbers: 15 = 6 + 6 + 3 = 15 + 0 + 0.
60 is a term because there are exactly 2 ways to express 60 as the sum of 3 triangular numbers: 60 = 36 + 21 + 3 = 45 + 15 + 0.
12 can be expressed as the sum of 3 triangular numbers in 3 ways, so it is not a term: 12 = 10 + 1 + 1 = 6 + 6 + 0 = 6 + 3 + 3. (End)

Cf. A000217, A060773, A061262, A002636.

Cf. A064825, A002097, A008443, A064816.

With[{max = 20}, t = Accumulate[Range[0, max]]; Select[Range[t[[-1]]], Length[IntegerPartitions[#, {3}, t]] == 2 &]] (* Amiram Eldar, May 14 2025 *)

for(n=1,150,if(sum(i=0,n,sum(j=0,i,sum(k=0,j,if(i*(i+1)/2+j*(j+1)/2+k*(k+1)/2-n,0,1))))==2,print1(n,",")))

{n: A002636(n) =2}. - R. J. Mathar, May 26 2025

More terms from Vladeta Jovovic, Jun 07 2002

Removed keyword "more" because this is probably finite. - R. J. Mathar, May 26 2025

A330810 a(n) is the largest number that can be expressed as the sum of three triangular numbers in exactly n ways.

Original entry on oeis.org

53, 194, 470, 788, 1730, 2000, 2693, 4310, 6053, 6845, 10688, 11348, 13970, 12923, 20768, 17135, 27830, 26480, 36245, 31688, 37073, 39983, 57860, 46940, 49148, 68258, 62810, 66515, 76985, 73868, 82850, 123878, 87890, 119810, 111053, 118490, 118880, 119183

Offset: 1

Jon E. Schoenfield, Jan 01 2020

nonn

One or more of the three triangular numbers may be zeros. If it were required that the triangular numbers be positive, sequence A330811 would result.

Cf. A000217, A002097, A060773, A061262, A063993, A064825, A071530, A111638, A258257, A258258, A330811.

A330811 a(n) is the largest number that can be expressed as the sum of three positive triangular numbers in exactly n ways.

Original entry on oeis.org

29, 119, 335, 713, 1730, 1328, 3413, 3485, 4565, 6053, 6950, 10688, 11348, 13970, 16778, 20768, 18173, 36245, 26480, 27203, 37073, 35033, 39983, 57860, 46940, 49148, 68258, 62810, 66515, 76985, 73868, 123878, 103403, 87890, 119810, 111053, 118490, 118880

Offset: 0

Jon E. Schoenfield, Jan 01 2020

nonn

If the triangular numbers were not required to be positive, sequence A330810 would result.

Cf. A000217, A002097, A060773, A061262, A063993, A064825, A071530, A111638, A258257, A258258, A330810.

A176661 Partial sums of A061262.

Original entry on oeis.org

0, 3, 15, 36, 88, 145, 236, 357, 493, 704, 896, 1122, 1531, 1862, 2229, 2635, 3146, 3653, 4539, 5176, 5948, 6669, 7540, 8492, 9594, 10660, 11887, 13079, 14720, 16341, 17737, 19118, 20619, 22351, 24143, 26070, 28012, 30413, 33024, 35575, 37997

Offset: 0

Jonathan Vos Post, Apr 23 2010

nonn

Partial sums of smallest number representable as the sum of 3 triangular numbers in exactly n ways. The subsequence of triangular numbers in the partial sum begins: 3, 15, 36. The subsequence of primes in the partial sum begins: 3, 1531, 11887, 17737, 37997, 43441.

			a(13) = 0 + 3 + 12 + 21 + 52 + 57 + 91 + 121 + 136 + 211 + 192 + 226 + 409 = 1531 is prime.

Cf. A061262, A000217, A053614, A060773, A002636, A124978.

a(n) = SUM[i=0..n] A061262(i).

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A053614 Numbers that are not the sum of distinct triangular numbers.

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A061262 Smallest number representable as the sum of 3 triangular numbers in exactly n ways.

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A274046 a(n) is the smallest positive integer which can be represented as the sum of distinct positive triangular numbers in exactly n ways, or 0 if no such integer exists.

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A117048 Prime numbers that are expressible as the sum of two positive triangular numbers.

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A111638 Numbers having a unique partition into three positive triangular numbers.

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A071530 Numbers that are the sum of 3 triangular numbers in exactly 2 ways.

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A330810 a(n) is the largest number that can be expressed as the sum of three triangular numbers in exactly n ways.

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A330811 a(n) is the largest number that can be expressed as the sum of three positive triangular numbers in exactly n ways.

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A176661 Partial sums of A061262.

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