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A066615
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Numbers that are not the sum of two or three distinct primes.
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1
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OFFSET
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1,2
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COMMENTS
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Probably there are no further terms.
An outgrowth of Goldbach's conjecture. "[I]n a letter to L. Euler (1742), C. F. Goldbach [asserted] that 'every odd number greater than 6 is equal to the sum of three primes.' Euler replied that Goldbach's conjecture was equivalent to the statement that every even number equal to or greater than 4 is equal to the sum of two primes. Because proving the second implies the first, but not the converse, most attention has been focused on the second representation. However, whether the statement is true for all even integers is still unsettled. Nevertheless, it is supported by existing evidence. A Russian mathematician, I. M. Vinogradov, proved that all large odd integers are the sum of three primes. Surprisingly, his techniques involve extremely subtle use of the theory of complex variables; no one has been able to extend them in order to solve Goldbach's conjecture." Andrews.
"Every number greater than 17 is the sum of 3 integers greater than 1 which are relatively prime in pairs." - Wells.
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REFERENCES
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George E. Andrews, "Number Theory," Dover Publ. Inc., NY, 1994, page 111.
Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, MA, 1996, Chapter 12, Pages 236-257.
Mark Herkommer, "Number Theory, A Programmer's Guide," McGraw-Hill, NY, 1999, pages 359-362.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, "An Introduction to The Theory of Numbers," Fifth Edition, John Wiley & Sons, Inc. NY, 1991, page 2.
Wacław Sierpiński, "250 Problems in Elementary Number Theory," New York: American Elsevier, Warsaw, 1970, pp. 4, 38-39.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, 1997, page 76.
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LINKS
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Table of n, a(n) for n=1..7.
Index entries for sequences related to Goldbach conjecture
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MATHEMATICA
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a = Table[ Prime[n], {n, 1, 100}]; b = {0}; Do[ b = Append[b, a[[i]] + a[[j]]], {j, 2, 100}, {i, 1, j - 1}]; Union[b]; c = {0}; Do[ c = Append[c, a[[i]] + a[[j]] + a[[k]]], {k, 3, 100}, {j, 2, k - 1}, {i, 1, j - 1}]; Union[c]; Complement[ Table[n, {n, 1, 541} ], Union[b, c]]
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CROSSREFS
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Intersection of A166081 and A124868.
Sequence in context: A050886 A079310 A116853 * A133951 A166081 A111124
Adjacent sequences: A066612 A066613 A066614 * A066616 A066617 A066618
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KEYWORD
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fini,nonn
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AUTHOR
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Amarnath Murthy, Dec 24 2001
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EXTENSIONS
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Entry revised by Robert G. Wilson v, Dec 27 2001
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STATUS
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approved
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