%I A074894
%S A074894 3,6,27,486
%N A074894 Full list of counterexamples for the k=3 version of the malicious apprentice 
               problem.
%C A074894 This is the problem of the farmer's helper who, when asked to weigh n 
               bags of grain, does so k at a time and reports the resulting binomial(n,
               k) combined weights with no indication of the k-tuples that produced 
               them. The problem: is can the weights of the bags be recovered?
%C A074894 For k=3 the answer is Yes unless n is one of the four terms of this sequence. 
               For k=2 see A057716.
%C A074894 The old entry with this sequence number was a duplicate of A030109.
%C A074894 The following references also apply to the general case of the problem.
%D A074894 I. N. Baker, Solutions of the functional equation (f(x))^2-f(x^2)=h(x), 
               Canad. Math. Bull., 3 (1960) 113-120.
%D A074894 W. W. Rouse Ball, A Short Account of the History of Mathematics.
%D A074894 E. Bolker, The finite Radon transform, Contemp. Math., 63 (1987) 27-50.
%D A074894 J. Boman, E. Bolker and P. O'Neil, The combinatorial Radon transform 
               modulo the symmetric group, Adv. Appl. Math., 12 (1991) 400-411.
%D A074894 Boman, Jan and Linusson, Svante, Examples of non-uniqueness for the combinatorial 
               Radon transform modulo the symmetric group. Math. Scand. 78 (1996), 
               207-212.
%D A074894 John A. Ewell, On the determination of sets by sets of sums of fixed 
               order, Canad. J. Math., 20 (1968) 596-611.
%D A074894 B. Gordon, A. S. Fraenkel and E. G. Straus, On the determination of sets 
               by the sets of sums of a certain order, Pacific J. Math., 12 (1962) 
               187-196.
%D A074894 R. K. Guy, Unsolved Problems in Number Theory, C5.
%D A074894 Ross A. Honsberger, A gem from combinatorics, Bull. ICA, 1 (1991) 56-58.
%D A074894 J. Lambek and L. Moser, On some two way classifications of the integers, 
               Canad. Math. Bull., 2 (1959) 85-89.
%D A074894 B. Liu and X. Zhang, On harmonious labelings of graphs, Ars Combin., 
               36 (1993) 315-326.
%D A074894 L. Moser, Problem E1248, Amer. Math. Monthly, 64 (1957) 507.
%D A074894 J. Ossowski, On a problem of Galvin, Congressus Numerantium, 96 (1993) 
               65-74.
%D A074894 D. G. Rogers, A functional equation: solution to Problem 89-19*, SIAM 
               Review, 32 (1990) 684-686.
%D A074894 J. L. Selfridge and E. G. Straus, On the determination of numbers by 
               their sums of a fixed order, Pacific J. Math., 8 (1958) 847-856.
%D A074894 P. Winkler, Mathematical Mind-Benders, Peters, Wellesley, MA, 2007; see 
               p. 27.
%e A074894 For n=27 Boman and Linusson give five examples of which the simplest 
               is {-4,-1^{10},2^{16}} and its negative, where exponents denote repetitions. 
               For n=486 Boman and Linusson give {-7,-4^{56},-1^{231},2^{176},5^{22}} 
               and its negative.
%Y A074894 See A057716 for the case k=2.
%Y A074894 Sequence in context: A083695 A060170 A097678 this_sequence A083675 A085076 
               A076711
%Y A074894 Adjacent sequences: A074891 A074892 A074893 this_sequence A074895 A074896 
               A074897
%K A074894 nonn,fini,full
%O A074894 1,1
%A A074894 N. J. A. Sloane (njas(AT)research.att.com), based on email from R. K. 
               Guy, Oct 30 2008