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A053614 Numbers that are not the sum of distinct triangular numbers. 17
2, 5, 8, 12, 23, 33 (list; graph; refs; listen; history; text; internal format)



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


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".


Table of n, a(n) for n=1..6.


Complement of A061208.


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.


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 *)


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.

Sequence in context: A066614 A305440 A045746 * A004711 A291922 A000789

Adjacent sequences:  A053611 A053612 A053613 * A053615 A053616 A053617




Jud McCranie, Mar 19 2000


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



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Last modified January 20 12:47 EST 2019. Contains 319330 sequences. (Running on oeis4.)