The numbers that are not the sum of at most 4 tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., (sequence A000797 in the OEIS) of 241 terms, with 343,867 conjectured to be the last such number.[3]
Pollock octahedral numbers conjecture: Every positive integer is the sum of at most 7 octahedral numbers.
This conjecture has been proven for all but finitely many positive integers.[4]
Pollock cube numbers conjecture: Every positive integer is the sum of at most 9 cube numbers.
^Frederick Pollock (1850). "On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders". Abstracts of the Papers Communicated to the Royal Society of London. 5: 922–924. JSTOR111069.