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

2, 5, 8, 12, 23, 33

Offset: 1

Author

Jud McCranie, Mar 19 2000

Keywords

• fini
• full
• nonn

Comments

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

Examples

			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.
		

References

• 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".

Links

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

Crossrefs

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.

Programs

Mathematica

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 *)

Formula

Complement of A061208.

Extensions

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