A002244 -id:A002244 - 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

A307598 Number of partitions of n into 3 distinct positive triangular numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 1, 1, 0, 3, 0, 2, 1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 0, 4, 1, 2, 1, 1, 1, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 1, 3, 2, 2, 2, 2, 1, 3, 2, 0, 4, 1, 1, 5, 1, 3, 2, 2, 3, 2, 2, 1, 4, 1, 2, 4, 2, 2, 3, 2, 1, 3, 2, 4, 3, 3, 2, 2, 3, 1, 6

Offset: 0

Ilya Gutkovskiy, Apr 17 2019

nonn

The greedy inverse starts 0, 10, 19, 37, 52, 82, 109, 136, 241, 226, 217, 247, 364, 427, 457, 541, 532, 577, 637, 961, 721, 787, 1066, 1102, 1381, 1267, 1564, 1192, 1396, 1816, 1501, 1612, 1927, 1942, 2242, 1792, 2842, 2587, 2557, 2422, ... - R. J. Mathar, Apr 28 2020

			a(19) = 2 because we have [15, 3, 1] and [10, 6, 3].

Cf. A000217, A002244, A008443, A024940, A025442, A053604, A063993, A224326, A307597.

a(n) = [x^n y^3] Product_{k>=1} (1 + y*x^(k*(k+1)/2)).

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

A002243 Numbers that are not the sum of 3 distinct triangular numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 12, 15, 20, 23, 33, 78

Offset: 1

N. J. A. Sloane, Dan Hoey

Cf. A000217, A002244.

Complement[Range[100],Union[Total/@Subsets[Accumulate[Range[0,12]], {3}]]] (* Harvey P. Dale, Sep 02 2014 *)

is(n)=for(c=ceil((sqrt(24*n-15)+3)/6),(sqrt(8*n-7)-1)/2,my(t=n-c*(c+1)/2);for(b=sqrtint(t-1)+1,min((sqrt(8*n+1)-1)/2,c-1), if(ispolygonal(t-b*(b+1)/2,3), return(0))));1
select(is,[1..100]) \\ Charles R Greathouse IV, Nov 25 2014

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

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A307598 Number of partitions of n into 3 distinct positive triangular numbers.

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

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A002243 Numbers that are not the sum of 3 distinct triangular numbers.

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