{"trustable":true,"prependHtml":"\u003cscript\u003e\n window.katexOptions \u003d {\n delimiters: [\n {left: \u0027\\\\(\u0027, right: \u0027\\\\)\u0027, display: false},\n ]\n };\n\u003c/script\u003e\n","sections":[{"title":"","value":{"format":"HTML","content":"\u003cp\u003eGiven a sequence of consecutive integers n,n+1,n+2,...,m, an anti-prime sequence is a rearrangement of these integers so that each adjacent pair of integers sums to a composite (non-prime) number. For example, if n \u003d 1 and m \u003d 10, one such anti-prime sequence is 1,3,5,4,2,6,9,7,8,10. This is also the lexicographically first such sequence.\u003c/p\u003e\n\n\u003cp\u003eWe can extend the definition by defining a degree danti-prime sequence as one where all consecutive subsequences of length 2,3,...,\u003ci\u003ed\u003c/i\u003e sum to a composite number. The sequence above is a degree 2 anti-prime sequence, but not a degree 3, since the subsequence 5, 4, 2 sums to 11. The lexicographically .rst degree 3 anti-prime sequence for these numbers is 1,3,5,4,6,2,10,8,7,9.\u003c/p\u003e\n\n\u003cb\u003e\u003cp\u003eInput\u003c/p\u003e\u003c/b\u003e\n\n\u003cp\u003eInput will consist of multiple input sets. Each set will consist of three integers, n, m, and d on a single line. The values of n, m and d will satisfy 1 \u0026lt;\u003d n \u0026lt; m \u0026lt;\u003d 1000, and 2 \u0026lt;\u003d d \u0026lt;\u003d 10. The line 0 0 0 will indicate end of input and should not be processed.\u003c/p\u003e\n\n\u003cp\u003e\u003cb\u003eOutput\u003c/b\u003e\u003c/p\u003e\n\n\u003cp\u003eFor each input set, output a single line consisting of a comma-separated list of integers forming a degree danti-prime sequence (do not insert any spaces and do not split the output over multiple lines). In the case where more than one anti-prime sequence exists, print the lexicographically first one (i.e., output the one with the lowest first value; in case of a tie, the lowest second value, etc.). In the case where no anti-prime sequence exists, output\u003c/p\u003e\n\n\u003cp\u003eNo anti-prime sequence exists.\u003c/p\u003e\n\n\u003cp\u003e\u003cb\u003eSample Input\u003c/b\u003e\u003c/p\u003e\n\u003cp\u003e\n1 10 2\u003cbr\u003e\n1 10 3\u003cbr\u003e\n1 10 5\u003cbr\u003e\n40 60 7\u003cbr\u003e\n0 0 0\u003cbr\u003e\n\u003c/p\u003e\n\n\u003cb\u003e\u003cp\u003eSample Output\u003c/p\u003e\u003c/b\u003e\n\u003cp\u003e\n1,3,5,4,2,6,9,7,8,10\u003cbr\u003e\n1,3,5,4,6,2,10,8,7,9\u003cbr\u003e\nNo anti-prime sequence exists.\u003cbr\u003e\n40,41,43,42,44,46,45,47,48,50,55,53,52,60,56,49,51,59,58,57,54\u003cbr\u003e\n\u003c/p\u003e\n"}}]}