cp's OEIS Frontend

A066615 Numbers that are not the sum of two or three distinct primes.

%I A066615 #26 Oct 13 2015 12:37:20
%S A066615 1,2,3,4,6,11,17
%N A066615 Numbers that are not the sum of two or three distinct primes.
%C A066615 Probably there are no further terms.
%C A066615 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.
%C A066615 "Every number greater than 17 is the sum of 3 integers greater than 1 which are relatively prime in pairs." - Wells.
%D A066615 George E. Andrews, "Number Theory," Dover Publ. Inc., NY, 1994, page 111.
%D A066615 Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, MA, 1996, Chapter 12, Pages 236-257.
%D A066615 Mark Herkommer, "Number Theory, A Programmer's Guide," McGraw-Hill, NY, 1999, pages 359-362.
%D A066615 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.
%D A066615 Wacław Sierpiński, "250 Problems in Elementary Number Theory," New York: American Elsevier, Warsaw, 1970, pp. 4, 38-39.
%D A066615 David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, 1997, page 76.
%H A066615 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%t A066615 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]]
%Y A066615 Intersection of A166081 and A124868.
%K A066615 fini,nonn
%O A066615 1,2
%A A066615 _Amarnath Murthy_, Dec 24 2001
%E A066615 Entry revised by _Robert G. Wilson v_, Dec 27 2001